essay in math

Math Essay Ideas for Students: Exploring Mathematical Concepts

Are you a student who's been tasked with writing a math essay? Don't fret! While math may seem like an abstract and daunting subject, it's actually full of fascinating concepts waiting to be explored. In this article, we'll delve into some exciting math essay ideas that will not only pique your interest but also impress your teachers. So grab your pens and calculators, and let's dive into the world of mathematics!

• The Beauty of Fibonacci Sequence

Have you ever wondered why sunflowers, pinecones, and even galaxies exhibit a mesmerizing spiral pattern? It's all thanks to the Fibonacci sequence! Explore the origin, properties, and real-world applications of this remarkable mathematical sequence. Discuss how it manifests in nature, art, and even financial markets. Unveil the hidden beauty behind these numbers and show how they shape the world around us.

• The Mathematics of Music

Did you know that music and mathematics go hand in hand? Dive into the relationship between these two seemingly unrelated fields and develop your writing skills . Explore the connection between harmonics, frequencies, and mathematical ratios. Analyze how musical scales are constructed and why certain combinations of notes create pleasant melodies while others may sound dissonant. Explore the fascinating world where numbers and melodies intertwine.

• The Geometry of Architecture

Architects have been using mathematical principles for centuries to create awe-inspiring structures. Explore the geometric concepts that underpin iconic architectural designs. From the symmetry of the Parthenon to the intricate tessellations in Islamic art, mathematics plays a crucial role in creating visually stunning buildings. Discuss the mathematical principles architects employ and how they enhance the functionality and aesthetics of their designs.

• Fractals: Nature's Infinite Complexity

Step into the mesmerizing world of fractals, where infinite complexity arises from simple patterns. Did you know that the famous Mandelbrot set , a classic example of a fractal, has been studied extensively and generated using computers? In fact, it is estimated that the Mandelbrot set requires billions of calculations to generate just a single image! This showcases the computational power and mathematical precision involved in exploring the beauty of fractal geometry.

Explore the beauty and intricacy of fractal geometry, from the famous Mandelbrot set to the Sierpinski triangle. Discuss the self-similarity and infinite iteration that define fractals and how they can be found in natural phenomena such as coastlines, clouds, and even in the structure of our lungs. Examine how fractal mathematics is applied in computer graphics, art, and the study of chaotic systems. Let the captivating world of fractals unfold before your eyes.

• The Game Theory Revolution

Game theory isn't just about playing games; it's a powerful tool used in various fields, from economics to biology. Dive into the world of strategic decision-making and explore how game theory helps us understand human behavior and predict outcomes. Discuss in your essay classic games like The Prisoner's Dilemma and examine how mathematical models can shed light on complex social interactions. Explore the cutting-edge applications of game theory in diverse fields, such as cybersecurity and evolutionary biology. If you still have difficulties choosing an idea for a math essay, find a reliable expert online. Ask them to write me an essay or provide any other academic assistance with your math assignments.

• Chaos Theory and the Butterfly Effect

While writing an essay, explore the fascinating world of chaos theory and how small changes can lead to big consequences. Discuss the famous Butterfly Effect and how it exemplifies the sensitive dependence on initial conditions. Delve into the mathematical principles behind chaotic systems and their applications in weather forecasting, population dynamics, and cryptography. Unravel the hidden order within apparent randomness and showcase the far-reaching implications of chaos theory.

• The Mathematics Behind Cryptography

In an increasingly digital world, cryptography plays a vital role in ensuring secure communication and data protection. Did you know that the global cybersecurity market is projected to reach a staggering $248.26 billion by 2023? This statistic emphasizes the growing importance of cryptography in safeguarding sensitive information.

Explore the mathematical foundations of cryptography and how it allows for the creation of unbreakable codes and encryption algorithms. Discuss the concepts of prime numbers, modular arithmetic, and public-key cryptography. Delve into the fascinating history of cryptography, from ancient times to modern-day encryption methods. In your essay, highlight the importance of mathematics in safeguarding sensitive information and the ongoing challenges faced by cryptographers.

Writing a math essay doesn't have to be a daunting task. By choosing a captivating topic and exploring the various mathematical concepts, you can turn your essay into a fascinating journey of discovery. Whether you're uncovering the beauty of the Fibonacci sequence, exploring the mathematical underpinnings of music, or delving into the game theory revolution, there's a world of possibilities waiting to be explored. So embrace the power of mathematics and let your creativity shine through your words!

Remember, these are just a few math essay ideas to get you started. Feel free to explore other mathematical concepts that ignite your curiosity. The world of mathematics is vast, and each concept has its own unique story to tell. So go ahead, unleash your inner mathematician, and embark on an exciting journey through the captivating realm of mathematical ideas!

Tobi Columb, a math expert, is a dedicated educator and explorer. He is deeply fascinated by the infinite possibilities of mathematics. Tobi's mission is to equip his students with the tools needed to excel in the realm of numbers. He also advocates for the benefits of a gluten-free lifestyle for students and people of all ages. Join Tobi on his transformative journey of mathematical mastery and holistic well-being.

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Math Essay | Essay on Math for Students and Children in English

February 13, 2024 by Prasanna

Math Essay: Mathematics is generally defined as the science that deals with numbers. It involves operations among numbers, and it also helps you to calculate the product price, how many discounted prizes here, and If you good in maths so you can calculate very fast. Mathematicians and scientists rely on mathematics principles in their real-life to experiments with new things every day. Many students say that ” I hate mathematics ” and maths is a useless subject, but it is wrong because without mathematics your life is tough to survive. Math has its applications in every field.

You can also find more  Essay Writing  articles on events, persons, sports, technology and many more.

Long and Short Essays on Math for Students and Kids in English

We are presenting students with essay samples on an extended essay of 500 words and a short of 150 words on the topic of math for reference.

Long Essay on Math 500 Words in English

Long Essay on Math is usually given to classes 7, 8, 9, and 10.

Mathematics is one of the common subjects that we study since our childhood. It is generally used in our daily life. Every person needs to learn some basics of it. Even counting money also includes math. Every work is linked with math in some way or the other. A person who does math is called a Mathematician.

Mathematics can be divided into two parts. The first is Pure mathematics, and the second is Applied mathematics. In Pure mathematics, we need to study the basic concept and structures of mathematics. But, on the other side, Applied mathematics involves the application of mathematics to solve problems that arise in various areas,(e.g.), science, engineering, and so on.

One couldn’t imagine the world without math. Math makes our life systematic, and every invention involves math. No matter what action a person is doing, he should know some basic maths. Every profession involves maths. Our present-day world runs on computers, and even computer runs with the help of maths. Every development that happens requires math.

Mathematics has a wide range of applications in our daily life. Maths generally deals with numbers. There are various topics in math, such as trigonometry; integration; differentiation, etc. All the subjects such as physics; chemistry; economy; commerce involve maths in some way or the other. Math is also used to find the relation between two numbers, and math is considered to be one of the most challenging subjects to learn. Math includes various numbers, and many symbols are used to show the relation between two different numbers.

Math is complicated to learn, and one needs to focus and concentrate more. Math is logical sometimes, and the logic needs to be derived out. Maths make our life easier and more straightforward. Math is considered to be challenging because it consists of many formulas that have to be learned, and many symbols and each symbol generally has its significance.

Some of the advantages of Math in our daily life

• Managing Money: Counting money and calculating simple interest, compound interest includes the usage of mathematics. Profit and loss are also computed using maths. Anything related to maths contains maths.
• Cooking: Maths is even used in cooking as estimating the number of ingredients that have to be used is calculated in numbers. Proportions also include maths.
• Home modelling: Calculating the area is essential in the construction of the home or home modelling. The size is also measured using maths. Even heights are also measured using maths.
• Travelling: Distance between two places and time taken to travel also includes maths. The amount of time taken revolves around maths. Almost every work is related to maths in some way or another. Maths contains some conditions that need to be followed, and maths has several formulas that have to be learned to become a mathematician.

Short Essay on Math 150 Words in English

Short Essay on Math is usually given to classes 1, 2, 3, 4, 5, and 6.

Maths is generally defined as the science of numbers and the operations performed among them. It deals with both alphabets along with numbers and involves addition, subtraction, multiplication, division, comparison, etc. It is used in every field. Maths consists of finding a relation between numbers, calculating the distance between two places, counting money, calculating profit and loss.

It is of two types pure and applied. Pure math deals with the basic structure and concept of maths, whereas applied mathematics deals with how maths is used it involves the application of maths in our daily life. All the subjects include maths, and hence maths is considered to be one of the primary and joint issues which need to be learned by everyone. One couldn’t imagine their life using maths. It has made our experience easy and straightforward. It has prevented chaos in our daily life. Hence learning maths is mandatory for everyone.

10 Lines on Math in English

• Father of Mathematics was Archimedes.
• Hypatia is the first woman know to know to have taught mathematics.
• From 0-1000 ,letter “A” only appears first in 1,000 ( “one thousand “).
• Zero (0) is the only number that can not be represented by Roman numerals.
• The Sign plus (+) and Minus(-) were discovered in 1489 A.D.
• Do you know that a Baseball field is of the perfect shape of a Rhombus.
• Jiffy is considered to be a unit of time for 1/100th of a second.
• 14th March International Day of Mathematics.
• Most mathematics symbols weren’t invented until the 16th century.
• The symbols for the division is called an Obelus.

FAQ’s on Math Essay

Question 1. What is Mathematics in simple words?

Answer: Mathematics is the study of shapes, patterns, numbers, and more. It involves a comparison between two numbers and calculating the distance between two places.

Question 2. Do we need mathematics every day?

Answer: Yes, we need mathematics every day, from buying a product to sell anything you want. Maths is present in our daily life, and no matter what work we do, maths is involved, and the application of maths is current in our everyday life.

Question 3. Who was the No.1 Mathematicians in the world?

Answer: Isaac Newton, who was a profound mathematician, is considered to be one of the best mathematicians in the world.

Question 4. What are the applications of maths?

Answer: Maths have various applications in our daily life. Maths is present everywhere from counting money to the calculating distance between two places. We could find math applications around.

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Essay on Importance of Mathematics in our Daily Life in 100, 200, and 350 words.

• Updated on  
• Dec 22, 2023

Mathematics is one of the core aspects of education. Without mathematics, several subjects would cease to exist. It’s applied in the science fields of physics, chemistry, and even biology as well. In commerce accountancy, business statistics and analytics all revolve around mathematics. But what we fail to see is that not only in the field of education but our lives also revolve around it. There is a major role that mathematics plays in our lives. Regardless of where we are, or what we are doing, mathematics is forever persistent. Let’s see how maths is there in our lives via our blog essay on importance of mathematics in our daily life. 

Table of Contents

• 1 Essay on Importance of Mathematics in our Daily life in 100 words 
• 2 Essay on Importance of Mathematics in our Daily life in 200 words
• 3 Essay on Importance of Mathematics in our Daily Life in 350 words

Essay on Importance of Mathematics in our Daily life in 100 words 

Mathematics is a powerful aspect even in our day-to-day life. If you are a cook, the measurements of spices have mathematics in them. If you are a doctor, the composition of medicines that make you provide prescription is made by mathematics. Even if you are going out for just some groceries, the scale that is used for weighing them has maths, and the quantity like ‘dozen apples’ has maths in it. No matter the task, one way or another it revolves around mathematics. Everywhere we go, whatever we do, has maths in it. We just don’t realize that. Maybe from now on, we will, as mathematics is an important aspect of our daily life.

Also Read:- Importance of Internet

Essay on Importance of Mathematics in our Daily life in 200 words

Mathematics, as a subject, is one of the most important subjects in our lives. Irrespective of the field, mathematics is essential in it. Be it physics, chemistry, accounts, etc. mathematics is there. The use of mathematics proceeds in our daily life to a major extent. It will be correct to say that it has become a vital part of us. Imagining our lives without it would be like a boat without a sail. It will be a shock to know that we constantly use mathematics even without realising the same. 

From making instalments to dialling basic phone numbers it all revolves around mathematics. 

Let’s take an example from our daily life. In the scenario of going out shopping, we take an estimate of hours. Even while buying just simple groceries, we take into account the weight of vegetables for scaling, weighing them on the scale and then counting the cash to give to the cashier. We don’t even realise it and we are already counting numbers and doing calculations. 

Without mathematics and numbers, none of this would be possible.

Hence we can say that mathematics helps us make better choices, more calculated ones throughout our day and hence make our lives simpler. 

Also Read:-   My Aim in Life

Also Read: How to Prepare for UPSC in 6 Months?

Essay on Importance of Mathematics in our Daily Life in 350 words

Mathematics is what we call a backbone, a backbone of science. Without it, human life would be extremely difficult to imagine. We cannot live even a single day without making use of mathematics in our daily lives. Without mathematics, human progress would come to a halt. 

Maths helps us with our finances. It helps us calculate our daily, monthly as well as yearly expenses. It teaches us how to divide and prioritise our expenses. Its knowledge is essential for investing money too. We can only invest money in property, bank schemes, the stock market, mutual funds, etc. only when we calculate the figures. Let’s take an example from the basic routine of a day. Let’s assume we have to make tea for ourselves. Without mathematics, we wouldn’t be able to calculate how many teaspoons of sugar we need, how many cups of milk and water we have to put in, etc. and if these mentioned calculations aren’t made, how would one be able to prepare tea? 

In such a way, mathematics is used to decide the portions of food, ingredients, etc. Mathematics teaches us logical reasoning and helps us develop problem-solving skills. It also improves our analytical thinking and reasoning ability. To stay in shape, mathematics helps by calculating the number of calories and keeping the account of the same. It helps us in deciding the portion of our meals. It will be impossible to think of sports without mathematics. For instance, in cricket, run economy, run rate, strike rate, overs bowled, overs left, number of wickets, bowling average, etc. are calculated. It also helps in predicting the result of the match. When we are on the road and driving, mathetics help us keep account of our speeds, the distance we have travelled, the amount of fuel left, when should we refuel our vehicles, etc. 

We can go on and on about how mathematics is involved in our daily lives. In conclusion, we can say that the universe revolves around mathematics. It encompasses everything and without it, we cannot imagine our lives. 

Also Read:- Essay on Pollution

Ans: Mathematics is a powerful aspect even in our day-to-day life. If you are a cook, the measurements of spices have mathematics in them. If you are a doctor, the composition of medicines that make you provide prescription is made by mathematics. Even if you are going out for just some groceries, the scale that is used for weighing them has maths, and the quantity like ‘dozen apples’ has maths in it. No matter the task, one way or another it revolves around mathematics. Everywhere we go, whatever we do, has maths in it. We just don’t realize that. Maybe from now on, we will, as mathematics is an important aspect of our daily life.

Ans: Mathematics, as a subject, is one of the most important subjects in our lives. Irrespective of the field, mathematics is essential in it. Be it physics, chemistry, accounts, etc. mathematics is there. The use of mathematics proceeds in our daily life to a major extent. It will be correct to say that it has become a vital part of us. Imagining our lives without it would be like a boat without a sail. It will be a shock to know that we constantly use mathematics even without realising the same.  From making instalments to dialling basic phone numbers it all revolves around mathematics. Let’s take an example from our daily life. In the scenario of going out shopping, we take an estimate of hours. Even while buying just simple groceries, we take into account the weight of vegetables for scaling, weighing them on the scale and then counting the cash to give to the cashier. We don’t even realise it and we are already counting numbers and doing calculations. Without mathematics and numbers, none of this would be possible. Hence we can say that mathematics helps us make better choices, more calculated ones throughout our day and hence make our lives simpler.  

Ans: Archimedes is considered the father of mathematics.

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Essays About Math: Top 10 Examples and Writing Prompts 

Love it or hate it, an understanding of math is said to be crucial to success. So, if you are writing essays about math, read our top essay examples.  

Mathematics is the study of numbers, shapes, and space using reason and usually a special system of symbols and rules for organizing them . It can be used for a variety of purposes, from calculating a business’s profit to estimating the mass of a black hole. However, it can be considered “controversial” to an extent.

Most students adore math or regard it as their least favorite. No other core subject has the same infamy as math for generating passionate reactions both for and against it. It has applications in every field, whether basic operations or complex calculus problems. Knowing the basics of math is necessary to do any work properly. 

If you are writing essays about Math, we have compiled some essay examples for you to get started. 

1. Mathematics: Problem Solving and Ideal Math Classroom by Darlene Gregory 

2. math essay by prasanna, 3. short essay on the importance of mathematics by jay prakash.

• 4.  Math Anxiety by Elias Wong

5. Why Math Isn’t as Useless as We Think by Murtaza Ali

1. mathematics – do you love or hate it, 2. why do many people despise math, 3. how does math prepare you for the future, 4. is mathematics an essential skill, 5. mathematics in the modern world.

“The trait of the teacher that is being strict is we know that will really help the students to change. But it will give a stress and pressure to students and that is one of the causes why students begin to dislike math. As a student I want a teacher that is not so much strict and giving considerations to his students. A teacher that is not giving loads of things to do and must know how to understand the reasons of his students.”

Gregory discusses the reasons for most students’ hatred of math and how teachers handle the subject in class. She says that math teachers do not explain the topics well, give too much work, and demand nothing less than perfection. To her, the ideal math class would involve teachers being more considerate and giving less work. 

You might also be interested in our ordinal number explainer.

“Math is complicated to learn, and one needs to focus and concentrate more. Math is logical sometimes, and the logic needs to be derived out. Maths make our life easier and more straightforward. Math is considered to be challenging because it consists of many formulas that have to be learned, and many symbols and each symbol generally has its significance.”

In her essay, Prasanna gives readers a basic idea of what math is and its importance. She additionally lists down some of the many uses of mathematics in different career paths, namely managing finances, cooking, home modeling and construction, and traveling. Math may seem “useless” and “annoying” to many, but the essay gives readers a clear message: we need math to succeed. 

“In this modern age of Science and Technology, emphasis is given on Science such as Physics, Chemistry, Biology, Medicine and Engineering. Mathematics, which is a Science by any criterion, also is an efficient and necessary tool being employed by all these Sciences. As a matter of fact, all these Sciences progress only with the aid of Mathematics. So it is aptly remarked, ‘Mathematics is a Science of all Sciences and art of all arts.’”

As its title suggests, Prakash’s essay briefly explains why math is vital to human nature. As the world continues to advance and modernize, society emphasizes sciences such as medicine, chemistry, and physics. All sciences employ math; it cannot be studied without math. It also helps us better our reasoning skills and maximizes the human mind. It is not only necessary but beneficial to our everyday lives. 

4.   Math Anxiety by Elias Wong

“Math anxiety affects different not only students but also people in different ways. It’s important to be familiar with the thoughts you have about yourself and the situation when you encounter math. If you are aware of unrealistic or irrational thoughts you can work to replace those thoughts with more positive and realistic ones.”

Wong writes about the phenomenon known as “math anxiety.” This term is used to describe many people’s hatred or fear of math- they feel that they are incapable of doing it. This anxiety is caused mainly by students’ negative experiences in math class, which makes them believe they cannot do well. Wong explains that some people have brains geared towards math and others do not, but this should not stop people from trying to overcome their math anxiety. Through review and practice of basic mathematical skills, students can overcome them and even excel at math. 

“We see that math is not an obscure subject reserved for some pretentious intellectual nobility. Though we may not be aware of it, mathematics is embedded into many different aspects of our lives and our world — and by understanding it deeply, we may just gain a greater understanding of ourselves.”

Similar to some of the previous essays, Ali’s essay explains the importance of math. Interestingly, he tells a story of the life of a person name Kyle. He goes through the typical stages of life and enjoys typical human hobbies, including Rubik’s cube solving. Throughout this “Kyle’s” entire life, he performed the role of a mathematician in various ways. Ali explains that math is much more prevalent in our lives than we think, and by understanding it, we can better understand ourselves. 

Writing Prompts on Essays about Math

Math is a controversial subject that many people either passionately adore or despise. In this essay, reflect on your feelings towards math, and state your position on the topic. Then, give insights and reasons as to why you feel this way. Perhaps this subject comes easily to you, or perhaps it’s a subject that you find pretty challenging. For an insightful and compelling essay, you can include personal anecdotes to relate to your argument. 

It is well-known that many people despise math. In this essay, discuss why so many people do not enjoy maths and struggle with this subject in school. For a compelling essay, gather interview data and statistics to support your arguments. You could include different sections correlating to why people do not enjoy this subject.

In this essay, begin by reading articles and essays about the importance of studying math. Then, write about the different ways that having proficient math skills can help you later in life. Next, use real-life examples of where maths is necessary, such as banking, shopping, planning holidays, and more! For an engaging essay, use some anecdotes from your experiences of using math in your daily life.

Many people have said that math is essential for the future and that you shouldn’t take a math class for granted. However, many also say that only a basic understanding of math is essential; the rest depends on one’s career. Is it essential to learn calculus and trigonometry? Choose your position and back up your claim with evidence. 

Prasanna’s essay lists down just a few applications math has in our daily lives. For this essay, you can choose any activity, whether running, painting, or playing video games, and explain how math is used there. Then, write about mathematical concepts related to your chosen activity and explain how they are used. Finally, be sure to link it back to the importance of math, as this is essentially the topic around which your essay is based. 

If you are interested in learning more, check out our essay writing tips !

For help with your essays, check out our round-up of the best essay checkers

260 Interesting Math Topics for Essays & Research Papers

Mathematics is the science of numbers and shapes. Writing about it can give you a fresh perspective and help to clarify difficult concepts. You can even use mathematical writing as a tool in problem-solving.

In this article, you will find plenty of interesting math topics. Besides, you will learn about branches of mathematics that you can choose from. And if the thought of letters and numbers makes your head swim, try our custom writing service . Our professionals will craft a paper for you in no time!

And now, let’s proceed to math essay topics and tips.

🔝 Top 10 Interesting Math Topics

✅ branches of mathematics, ✨ fun math topics.

• 🏫 Math Topics for High School
• 🎓 College Math Topics
• 🤔 Advanced Math
• 📚 Math Research
• ✏️ Math Education
• 💵 Business Math

🔍 References

• Number theory in everyday life.
• Logicist definitions of mathematics.
• Multivariable vs. vector calculus.
• 4 conditions of functional analysis.
• Random variable in probability theory.
• How is math used in cryptography?
• The purpose of homological algebra.
• Concave vs. convex in geometry.
• The philosophical problem of foundations.
• Is numerical analysis useful for machine learning?

What exactly is mathematics ? First and foremost, it is very old. Ancient Greeks and Persians were already utilizing mathematical tools. Nowadays, we consider it an interdisciplinary language.

Biologists, linguists, and sociologists alike use math in their work. And not only that, we all deal with it in our daily lives. For instance, it manifests in the measurement of time. We often need it to calculate how much our groceries cost and how much paint we need to buy to cover a wall.

Simply put, mathematics is a universal instrument for problem-solving. We can divide pure math into three branches: geometry, arithmetic, and algebra. Let’s take a closer look:

• Geometry By studying geometry, we try to comprehend our physical surroundings. Geometric shapes can be simple, like a triangle. Or, they can form complicated figures, like a rhombicosidodecahedron.
• Arithmetic Arithmetic deals with numbers and simple operations: subtraction, addition, division, and multiplication.
• Algebra Algebra is used when the exact numbers are unclear. Instead, they are replaced with letters. Businesses often need algebra to predict their sales.

It’s true that most high school students don’t like math. However, that doesn’t mean it can’t be a fun and compelling subject. In the following section, you will find plenty of enthralling mathematical topics for your paper.

If you’re struggling to start working on your essay, we have some fun and cool math topics to offer. They will definitely engage you and make the writing process enjoyable. Besides, fun math topics can show everyone that even math can be entertaining or even a bit silly.

• The link between mathematics and art – analyzing the Golden Ratio in Renaissance-era paintings.
• An evaluation of Georg Cantor’s set theory.
• The best approaches to learning math facts and developing number sense.
• Different approaches to probability as explored through analyzing card tricks. 
• Chess and checkers – the use of mathematics in recreational activities.
• The five types of math used in computer science.
• Real-life applications of the Pythagorean Theorem. 
• A study of the different theories of mathematical logic.
• The use of game theory in social science.
• Mathematical definitions of infinity and how to measure it.
• What is the logic behind unsolvable math problems?
• An explanation of mean, mode, and median using classroom math grades.
• The properties and geometry of a Möbius strip.
• Using truth tables to present the logical validity of a propositional expression.
• The relationship between Pascal’s Triangle and The Binomial Theorem. 
• The use of different number types: the history.
• The application of differential geometry in modern architecture.
• A mathematical approach to the solution of a Rubik’s Cube.
• Comparison of predictive and prescriptive statistical analyses.
• Explaining the iterations of the Koch snowflake.
• The importance of limits in calculus.
• Hexagons as the most balanced shape in the universe.
• The emergence of patterns in chaos theory.
• What were Euclid’s contributions to the field of mathematics?
• The difference between universal algebra and abstract algebra.

🏫 Math Essay Topics for High School

When writing a math paper, you want to demonstrate that you understand a concept. It can be helpful if you need to prepare for an exam. Choose a topic from this section and decide what you want to discuss.

• Explain what we need Pythagoras’ theorem for.
• What is a hyperbola?
• Describe the difference between algebra and arithmetic.
• When is it unnecessary to use a calculator ?
• Find a connection between math and the arts.
• How do you solve a linear equation?
• Discuss how to determine the probability of rolling two dice.
• Is there a link between philosophy and math?
• What types of math do you use in your everyday life?
• What is the numerical data?
• Explain how to use the binomial theorem.
• What is the distributive property of multiplication?
• Discuss the major concepts in ancient Egyptian mathematics. 
• Why do so many students dislike math?
• Should math be required in school?
• How do you do an equivalent transformation?
• Why do we need imaginary numbers?
• How can you calculate the slope of a curve?
• What is the difference between sine, cosine, and tangent?
• How do you define the cross product of two vectors?
• What do we use differential equations for?
• Investigate how to calculate the mean value.
• Define linear growth.
• Give examples of different number types.
• How can you solve a matrix?

🎓 College Math Topics for a Paper

Sometimes you need more than just formulas to explain a complex idea. That’s why knowing how to express yourself is crucial. It is especially true for college-level mathematics. Consider the following ideas for your next research project:

• What do we need n-dimensional spaces for?
• Explain how card counting works.
• Discuss the difference between a discrete and a continuous probability distribution. 
• How does encryption work? 
• Describe extremal problems in discrete geometry.
• What can make a math problem unsolvable?
• Examine the topology of a Möbius strip.

• What is K-theory? 
• Discuss the core problems of computational geometry.
• Explain the use of set theory .
• What do we need Boolean functions for?
• Describe the main topological concepts in modern mathematics.
• Investigate the properties of a rotation matrix.
• Analyze the practical applications of game theory.
• How can you solve a Rubik’s cube mathematically?
• Explain the math behind the Koch snowflake.
• Describe the paradox of Gabriel’s Horn.
• How do fractals form?
• Find a way to solve Sudoku using math.
• Why is the Riemann hypothesis still unsolved?
• Discuss the Millennium Prize Problems.
• How can you divide complex numbers?
• Analyze the degrees in polynomial functions.
• What are the most important concepts in number theory?
• Compare the different types of statistical methods.

🤔 Advanced Topics in Math to Write a Paper on

Once you have passed the trials of basic math, you can move on to the advanced section. This area includes topology, combinatorics, logic, and computational mathematics. Check out the list below for enticing topics to write about:

• What is an abelian group?
• Explain the orbit-stabilizer theorem.
• Discuss what makes the Burnside problem influential.
• What fundamental properties do holomorphic functions have?
• How does Cauchy’s integral theorem lead to Cauchy’s integral formula?
• How do the two Picard theorems relate to each other?
• When is a trigonometric series called a Fourier series?
• Give an example of an algorithm used for machine learning.
• Compare the different types of knapsack problems.
• What is the minimum overlap problem?
• Describe the Bernoulli scheme.
• Give a formal definition of the Chinese restaurant process.
• Discuss the logistic map in relation to chaos.
• What do we need the Feigenbaum constants for?
• Define a difference equation.
• Explain the uses of the Fibonacci sequence.
• What is an oblivious transfer?
• Compare the Riemann and the Ruelle zeta functions.
• How can you use elementary embeddings in model theory?
• Analyze the problem with the wholeness axiom and Kunen’s inconsistency theorem.
• How is Lie algebra used in physics ?
• Define various cases of algebraic cycles.
• Why do we need étale cohomology groups to calculate algebraic curves?
• What does non-Euclidean geometry consist of?
• How can two lines be ultraparallel?

📚 Math Research Topics for a Paper

Choosing the right topic is crucial for a successful research paper in math. It should be hard enough to be compelling, but not exceeding your level of competence. If possible, stick to your area of knowledge. This way your task will become more manageable. Here are some ideas:

• Write about the history of calculus.
• Why are unsolved math problems significant?
• Find reasons for the gender gap in math students.
• What are the toughest mathematical questions asked today?
• Examine the notion of operator spaces.
• How can we design a train schedule for a whole country?
• What makes a number big?

• How can infinities have various sizes?
• What is the best mathematical strategy to win a game of Go?
• Analyze natural occurrences of random walks in biology.
• Explain what kind of mathematics was used in ancient Persia.
• Discuss how the Iwasawa theory relates to modular forms.
• What role do prime numbers play in encryption?
• How did the study of mathematics evolve?
• Investigate the different Tower of Hanoi solutions.
• Research Napier’s bones. How can you use them?
• What is the best mathematical way to find someone who is lost in a maze?
• Examine the Traveling Salesman Problem. Can you find a new strategy?
• Describe how barcodes function.
• Study some real-life examples of chaos theory. How do you define them mathematically?
• Compare the impact of various ground-breaking mathematical equations .
• Research the Seven Bridges of Königsberg. Relate the problem to the city of your choice.
• Discuss Fisher’s fundamental theorem of natural selection.
• How does quantum computing work?
• Pick an unsolved math problem and say what makes it so difficult.

✏️ Math Education Research Topics

For many teachers, the hardest part is to keep the students interested. When it comes to math, it can be especially challenging. It’s crucial to make complicated concepts easy to understand. That’s why we need research on math education.

• Compare traditional methods of teaching math with unconventional ones.
• How can you improve mathematical education in the U.S.?
• Describe ways of encouraging girls to pursue careers in STEM fields.
• Should computer programming be taught in high school?
• Define the goals of mathematics education .
• Research how to make math more accessible to students with learning disabilities. 
• At what age should children begin to practice simple equations?
• Investigate the effectiveness of gamification in algebra classes. 
• What do students gain from taking part in mathematics competitions?
• What are the benefits of moving away from standardized testing ?
• Describe the causes of “ math anxiety .” How can you overcome it?
• Explain the social and political relevance of mathematics education.
• Define the most significant issues in public school math teaching.
• What is the best way to get children interested in geometry?
• How can students hone their mathematical thinking outside the classroom?
• Discuss the benefits of using technology in math class. 
• In what way does culture influence your mathematical education?
• Explore the history of teaching algebra.
• Compare math education in various countries.

• How does dyscalculia affect a student’s daily life?
• Into which school subjects can math be integrated?
• Has a mathematics degree increased in value over the last few years?
• What are the disadvantages of the Common Core Standards?
• What are the advantages of following an integrated curriculum in math?
• Discuss the benefits of Mathcamp.

🧮 Algebra Topics for a Paper

The elegance of algebra stems from its simplicity. It gives us the ability to express complex problems in short equations. The world was changed forever when Einstein wrote down the simple formula E=mc². Now, if your algebra seminar requires you to write a paper, look no further! Here are some brilliant prompts:

• Give an example of an induction proof.
• What are F-algebras used for?
• What are number problems?
• Show the importance of abstract algebraic thinking. 
• Investigate the peculiarities of Fermat’s last theorem.
• What are the essentials of Boolean algebra?
• Explore the relationship between algebra and geometry.
• Compare the differences between commutative and noncommutative algebra.
• Why is Brun’s constant relevant?
• How do you factor quadratics?
• Explain Descartes’ Rule of Signs.
• What is the quadratic formula?
• Compare the four types of sequences and define them.
• Explain how partial fractions work.
• What are logarithms used for?
• Describe the Gaussian elimination.
• What does Cramer’s rule state?
• Explore the difference between eigenvectors and eigenvalues.
• Analyze the Gram-Schmidt process in two dimensions.
• Explain what is meant by “range” and “domain” in algebra.
• What can you do with determinants?
• Learn about the origin of the distance formula.
• Find the best way to solve math word problems.
• Compare the relationships between different systems of equations.
• Explore how the Rubik’s cube relates to group theory.

📏 Geometry Topics for a Research Paper

Shapes and space are the two staples of geometry. Since its appearance in ancient times, it has evolved into a major field of study. Geometry’s most recent addition, topology, explores what happens to an object if you stretch, shrink, and fold it. Things can get pretty crazy from here! The following list contains 25 interesting geometry topics:

• What are the Archimedean solids?
• Find real-life uses for a rhombicosidodecahedron.
• What is studied in projective geometry?
• Compare the most common types of transformations.
• Explain how acute square triangulation works.
• Discuss the Borromean ring configuration.
• Investigate the solutions to Buffon’s needle problem.
• What is unique about right triangles?

• Describe the notion of Dirac manifolds.
• Compare the various relationships between lines.
• What is the Klein bottle?
• How does geometry translate into other disciplines, such as chemistry and physics?
• Explore Riemannian manifolds in Euclidean space.
• How can you prove the angle bisector theorem?
• Do a research on M.C. Escher’s use of geometry.
• Find applications for the golden ratio .
• Describe the importance of circles.
• Investigate what the ancient Greeks knew about geometry.
• What does congruency mean?
• Study the uses of Euler’s formula.
• How do CT scans relate to geometry?
• Why do we need n-dimensional vectors?
• How can you solve Heesch’s problem?
• What are hypercubes?
• Analyze the use of geometry in Picasso’s paintings.

➗ Calculus Topics to Write a Paper on

You can describe calculus as a more complicated algebra. It’s a study of change over time that provides useful insights into everyday problems. Applied calculus is required in a variety of fields such as sociology, engineering, or business. Consult this list of compelling topics on a calculus paper:

• What are the differences between trigonometry, algebra, and calculus?
• Explain the concept of limits.
• Describe the standard formulas needed for derivatives.
• How can you find critical points in a graph?
• Evaluate the application of L’Hôpital’s rule.
• How do you define the area between curves?
• What is the foundation of calculus?

• How does multivariate calculus work?
• Discuss the use of Stokes’ theorem.
• What does Leibniz’s integral rule state?
• What is the Itô stochastic integral?
• Explore the influence of nonstandard analysis on probability theory.
• Research the origins of calculus.
• Who was Maria Gaetana Agnesi?
• Define a continuous function.
• What is the fundamental theorem of calculus?
• How do you calculate the Taylor series of a function?
• Discuss the ways to resolve Runge’s phenomenon.
• Explain the extreme value theorem.
• What do we need predicate calculus for?
• What are linear approximations?
• When does an integral become improper?
• Describe the Ratio and Root Tests.
• How does the method of rings work?
• Where do we apply calculus in real-life situations?

💵 Business Math Topics to Write About

You don’t have to own a company to appreciate business math. Its topics range from credits and loans to insurance, taxes, and investment. Even if you’re not a mathematician, you can use it to handle your finances. Sounds interesting? Then have a look at the following list:

• What are the essential skills needed for business math?
• How do you calculate interest rates?
• Compare business and consumer math.
• What is a discount factor?
• How do you know that an investment is reasonable?
• When does it make sense to pay a loan with another loan?
• Find useful financing techniques that everyone can use.
• How does critical path analysis work?
• Explain how loans work.
• Which areas of work utilize operations research?
• How do businesses use statistics?
• What is the economic lot scheduling problem?
• Compare the uses of different chart types.
• What causes a stock market crash?
• How can you calculate the net present value?
• Explore the history of revenue management.
• When do you use multi-period models?
• Explain the consequences of depreciation.
• Are annuities a good investment?
• Would the U.S. financially benefit from discontinuing the penny?
• What caused the United States housing crash in 2008?
• How do you calculate sales tax?
• Describe the notions of markups and markdowns. 
• Investigate the math behind debt amortization.
• What is the difference between a loan and a mortgage?

With all these ideas, you are perfectly equipped for your next math paper. Good luck!

• What Is Calculus?: Southern State Community College
• What Is Mathematics?: Tennessee Tech University
• What Is Geometry?: University of Waterloo
• What Is Algebra?: BBC
• Ten Simple Rules for Mathematical Writing: Ohio State University
• Practical Algebra Lessons: Purplemath
• Topics in Geometry: Massachusetts Institute of Technology
• The Geometry Junkyard: All Topics: Donald Bren School of Information and Computer Sciences
• Calculus I: Lamar University
• Business Math for Financial Management: The Balance Small Business
• What Is Mathematics: Life Science
• What Is Mathematics Education?: University of California, Berkeley
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I need a writer on algebra. I am a PhD student.Can i be helped by anybody/expert?

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ESSAY SAUCE

FOR STUDENTS : ALL THE INGREDIENTS OF A GOOD ESSAY

• Mathematics essays

Our free mathematics essay examples include popular topics such as algorithms, applied mathematics, calculus, knot theory, linear algebra, and more.

Euler’s identity

Euler’s identity is an equality found in mathematics that has been compared to a Shakespearean sonnet and described as “the most beautiful equation.” It is a special case of a foundational equation in complex arithmetic called Euler’s Formula, which the late great physicist Richard Feynman called in his lectures “our jewel” and “the most remarkable formula in … Read more

The Banach-Tarski paradox

Mathematics is seen by many as a mysterious and often unsettling subject. Answers often hide behind layers and layers of complicated equations, formulas and ciphers, the application of advanced concepts to real life is limited and I often find myself more confused after class than when I first entered. However, the real beauty of Mathematics … Read more

Pascal’s triangle, binomial theorem

What is Pascal’s Triangle? Pascal’s Triangle was named after Blaise Pascal. Pascal’s triangle starts with the number 1 and goes down the scale. When you start with one, add more numbers in a triangular shape, like a pyramid of some sort. All the numbers on the surrounding right and left sides of the triangle are … Read more

My journey of teaching and learning mathematics since embarking on a PGCE Mathematics course

Before I came to study a Post Graduate Certificate (PGCE) Mathematics course at University College London Institute of Education (UCL IOE), I had been working as an Academic Tutor at a behavioural centre, linked to a mainstream secondary school for the past 7 months. Students placed here had either learning difficulties or behaviour issues experienced … Read more

Aircraft – mathematics

Math SL Internal Assessment Lift and Drag Introduction When you look at aircrafts, they look like they shouldn’t be able to leave the ground because of how big they are. I always watched aircrafts, take off and land, over and over. According to Newton’s Third Law, every action has an equal and opposite reaction, lift … Read more

The World of Mathematics

Mathematics is often considered a useless discipline because people think they do not use advanced math in their life. It is a misunderstanding. Not using advanced math does not mean it is not vital. If you trade stocks, you will read many stock analysis reports. These reports use mathematical knowledge. Many students in the United … Read more

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Mathematics introduction

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Niels Henrik Abel

For over two centuries, mathematicians had trouble in finding a solution to the quintic equation, that is until Niels Henrik Abel formulated a theory. Abel was a Norwegian mathematician born on August 5, 1802, and his talent and potential in the field of mathematics was already present at a young age, leading him to become … Read more

The narrative of zero

Numbers surround us. They stamp our days, light our evenings, foresee our climate, and keep us on course. They drive business and support human progress. The beginning of the numerals makes disarray between the historical backdrop of mathematics and the historical backdrop of our modern numerals. The narrative of zero alludes to something can be … Read more

Emmy Noether

She was more than a mathematician to the people she met and to the people she inspired. She even has managed to inspire people long after she has passed. Emmy Noether was born on March 23, 1882 in Bavaria Germany. Growing up she wanted to go to college but back then women weren’t allowed to … Read more

Bitopological Approximation Space with Application to Data Reduction in Multivalued Information Systems

Abstract: In this work we generalize Pawlak approximation space to bitopological approximation space. One binary relation can define two subbases of two topological spaces. Membership, equillity and inclusion relations using rough approximations are defined and studied in bitopological aapproximation space. Some new measures that measure the accuracy and the quality of approximations are defined and … Read more

Statistics overview

Statistics is a form of mathematical analysis that uses quantified models, representations and synopses for a given set of experimental data or real life studies. Statistical analysis involves the process of gathering and evaluating data and then summarizing the data into a mathematical form. Statistics is a term used to summarize a process that a … Read more

Numerical Weather Prediction

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The Mathematics of Our Universe

Abstract In this report, we start by defining key aspects of classical Lagrangian mechanics including the principle of least action and how one can use this to derive the Euler-Lagrange equations. Momentum and Conservation laws shall also be introduced, deriving relations between position, momenta and the Lagrangian of a given system. Following this, we develop … Read more

Ideas for your next mathematics essay

Stuck for a title for your next essay? Here are some ideas to inspire you:

• The Mathematics of Music: Exploring the Relationship between Mathematics and Music – This essay would examine the connections between music and mathematics, including the use of mathematical concepts in musical composition and the study of the mathematics of sound.
• The Golden Ratio: A Mathematical and Aesthetic Marvel – This essay would discuss the concept of the golden ratio and its applications in art, architecture, and design. It would explore the beauty and symmetry of this mathematical principle.
• Mathematics in Sports: Analyzing the Numbers Behind Athletic Performance – This essay would explore the use of mathematics in sports, including the use of statistics and analytics to analyze athletic performance and predict outcomes.
• Chaos Theory: The Science of Nonlinear Systems – This essay would discuss the concept of chaos theory and its applications in various fields, such as meteorology, physics, and economics. It would explore the idea that small changes in initial conditions can have a significant impact on the final outcome of a system.
• The Mathematics of Cryptography: Securing Information in the Digital Age – This essay would examine the use of mathematics in cryptography, including the principles of encryption and decryption, and how these concepts are applied to secure information in the digital age.
• Fractals: The Beauty of Infinite Complexity – This essay would explore the concept of fractals and their applications in art, nature, and science. It would discuss the beauty and complexity of these repeating patterns found in nature and how they are used in various fields of study.
• Mathematical Models in Biology: Understanding the Complexities of Life – This essay would discuss the use of mathematical models in biology, including the modeling of population growth, the spread of disease, and the behavior of organisms. It would explore how these models help scientists understand the complex systems that make up living organisms.
• The Mathematics of Finance: Analyzing Investments and Markets – This essay would examine the use of mathematics in finance, including the principles of financial analysis, investments, and risk management. It would explore how mathematics is used to understand and predict market trends.
• Geometry in Art: The Intersection of Math and Creativity – This essay would discuss the use of geometry in art, including the use of shapes, patterns, and symmetry. It would explore how artists use mathematical concepts to create beautiful and compelling works of art.
• The History of Mathematics: From Ancient Times to Modern-Day Advances – This essay would trace the history of mathematics, from its origins in ancient civilizations to modern-day advancements in the field. It would explore the contributions of key mathematicians throughout history and the evolution of mathematical concepts and principles over time.

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Extended Essay: Group 5: Mathematics

• General Timeline
• Group 1: English Language and Literature
• Group 2: Language Acquisition
• Group 3: Individuals and Societies
• Group 4: Sciences
• Group 5: Mathematics
• Group 6: The Arts
• Interdisciplinary essays
• Six sub-categories for WSEE
• IB Interdisciplinary EE Assessment Guide
• Brainstorming
• Pre-Writing
• Research Techniques
• The Research Question
• Paraphrasing, Summarising and Quotations
• Writing an EE Introduction
• Writing the main body of your EE
• Writing your EE Conclusion
• Sources: Finding, Organising and Evaluating Them
• Conducting Interviews and Surveys
• Citing and Referencing
• Check-in Sessions
• First Formal Reflection
• Second Formal Reflection
• Final Reflection (Viva Voce)
• Researcher's Reflection Space (RRS) Examples
• Information for Supervisors
• How is the EE Graded?
• EE Online Resources
• Stavanger Public Library
• Exemplar Essays
• Extended Essay Presentations
• ISS High School Academic Honesty Policy

Mathematics

An extended essay (EE) in mathematics is intended for students who are writing on any topic that has a mathematical focus and it need not be confined to the theory of mathematics itself.

Essays in this group are divided into six categories:

• the applicability of mathematics to solve both real and abstract problems
• the beauty of mathematics—eg geometry or fractal theory
• the elegance of mathematics in the proving of theorems—eg number theory
• the history of mathematics: the origin and subsequent development of a branch of mathematics over a period of time, measured in tens, hundreds or thousands of years
• the effect of technology on mathematics:
• in forging links between different branches of mathematics,
• or in bringing about a new branch of mathematics, or causing a particular branch to flourish.

These are just some of the many different ways that mathematics can be enjoyable or useful, or, as in many cases, both.

For an Introduction in a Mathematics EE look HERE . 

Choice of topic

The EE may be written on any topic that has a mathematical focus and it need not be confined to the theory of mathematics itself.

Students may choose mathematical topics from fields such as engineering, the sciences or the social sciences, as well as from mathematics itself.

Statistical analyses of experimental results taken from other subject areas are also acceptable, provided that they focus on the modeling process and discuss the limitations of the results; such essays should not include extensive non-mathematical detail.

A topic selected from the history of mathematics may also be appropriate, provided that a clear line of mathematical development is demonstrated. Concentration on the lives of, or personal rivalries between, mathematicians would be irrelevant and would not score highly on the assessment criteria.

It should be noted that the assessment criteria give credit for the nature of the investigation and for the extent that reasoned arguments are applied to an appropriate research question.

Students should avoid choosing a topic that gives rise to a trivial research question or one that is not sufficiently focused to allow appropriate treatment within the requirements of the EE.

Students will normally be expected either to extend their knowledge beyond that encountered in the Diploma Programme mathematics course they are studying or to apply techniques used in their mathematics course to modeling in an appropriately chosen topic.

However, it is very important to remember that it is an essay that is being written, not a research paper for a journal of advanced mathematics, and no result, however impressive, should be quoted without evidence of the student’s real understanding of it.

Example and Treatment of Topic

Examples of topics

These examples are just for guidance. Students must ensure their choice of topic is focused (left-hand column) rather than broad (right-hand column

Treatment of the topic

Whatever the title of the EE, students must apply good mathematical practice that is relevant to the

chosen topic, including:

• data analysed using appropriate techniques

• arguments correctly reasoned

• situations modeled using correct methodology

• problems clearly stated and techniques at the correct level of sophistication applied to their solution.

Research methods

Students must be advised that mathematical research is a long-term and open-ended exploration of a set of related mathematical problems that are based on personal observations. 

The answers to these problems connect to and build upon each other over time.

Students’ research should be guided by analysis of primary and secondary sources.

A primary source for research in mathematics involves:

• data-gathering

• visualization

• abstraction

• conjecturing

• proof.

A secondary source of research refers to a comprehensive review of scholarly work, including books, journal articles or essays in an edited collection.

A literature review for mathematics might not be as extensive as in other subjects, but students are expected to demonstrate their knowledge and understanding of the mathematics they are using in the context of the broader discipline, for example how the mathematics they are using has been applied before, or in a different area to the one they are investigating.

Writing the essay

Throughout the EE students should communicate mathematically:

• describing their way of thinking

• writing definitions and conjectures

• using symbols, theorems, graphs and diagrams

• justifying their conclusions.

There must be sufficient explanation and commentary throughout the essay to ensure that the reader does not lose sight of its purpose in a mass of mathematical symbols, formulae and analysis.

The unique disciplines of mathematics must be respected throughout. Relevant graphs and diagrams are often important and should be incorporated in the body of the essay, not relegated to an appendix.

However, lengthy printouts, tables of results and computer programs should not be allowed to interrupt the development of the essay, and should appear separately as footnotes or in an appendix. Proofs of key results may be included, but proofs of standard results should be either omitted or, if they illustrate an important point, included in an appendix.

Examples of topics, research questions and suggested approaches

Once students have identified their topic and written their research question, they can decide how to

research their answer. They may find it helpful to write a statement outlining their broad approach. These

examples are for guidance only.

An important note on “double-dipping”

Students must ensure that their EE does not duplicate other work they are submitting for the Diploma Programme. For example, students are not permitted to repeat any of the mathematics in their IA in their EE, or vice versa.

The mathematics EE and internal assessment

An EE in mathematics is not an extension of the internal assessment (IA) task. Students must ensure that they understand the differences between the two.

• The EE is a more substantial piece of work that requires formal research
• The IA is an exploration of an idea in mathematics.

It is not appropriate for a student to choose the same topic for an EE as the IA. There would be too much danger of duplication and it must therefore be discouraged.

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How to write a good mathematical paper?

I hesitate to ask this question. However I read many advices from math.stackexchange, and I couldn't find anything similar.

A good time always goes too fast! Two years are fled. In the third year of PHD, my major is general topology and I'm facing with graduation from PHD. I do enjoy research, however the pressure to publish makes me be agitated and not quite, for I haven't publish any paper. I find, sometimes, doing research and to publish are contradictory.

Here is my question: How to write a good mathematical paper? Could anybody give me some suggestions?

Thanks ahead.

• soft-question
• article-writing

• 10 $\begingroup$ It appears that you have done research and produced some results. Writing a paper should be easy once you have something to write about. You should write done your results and proofs as clearly as possible. Try not to get bogged down by details, and consult your advisor to determine what details an expert should be able fill in themselves. Check your spelling and grammar. Talk to advisor about the known journals of your area and those likely to accept your paper. The important thing is to consult your advisor, but since you have results, I think the hardest part is already done. $\endgroup$ –  William Commented Aug 11, 2012 at 1:38
• 4 $\begingroup$ Don't do the classic textbook approach of having your main statement and the preceeding lemmas be a total secret until the reader get to the specific page. The most negative comment I recieved on the only thing I have ever written close to a mathematical paper was that I started out too "heavy". I was told it was better to have an abstract (which should only be a sentence or two) and then a relatively short section explaining basic ideas in a way that wouldn't be considered a wall of text or an overload of definitions and constructions. $\endgroup$ –  Arthur Commented Aug 11, 2012 at 1:48
• 12 $\begingroup$ You have to have a result first. $\endgroup$ –  user2468 Commented Aug 11, 2012 at 3:44
• 3 $\begingroup$ @Paul Unfortunately I cannot edit it, although it does work for me. Here is the direct link: terrytao.wordpress.com/advice-on-writing-papers $\endgroup$ –  Sniper Clown Commented Aug 11, 2012 at 6:58
• 9 $\begingroup$ Having just refereed my first paper, I'll try to say a few of meaningful things. (1) Don't obfuscate with formally correct notation where a general idea -- simply expressible in English with perhaps a few mathematical symbols -- will suffice. (2) Be consistent with notations/conventions. (3) If your proof involves a long, tedious, technical component, break it up into segments and explain what it is you are attempting to do in each segment. (4) Remember that while after two years of intense study everything seems natural and clear, it might not be for someone seeing it for the first time. $\endgroup$ –  user642796 Commented Aug 11, 2012 at 7:07

5 Answers 5

As someone who is currently working on my first mathematical paper, I've found this guide from MIT to be very helpful. It covers both writing a clear and precise paper in general as well as the specific challenges presented by a mathematical paper. It's also fun to read! For example, the author likes to illustrate common mistakes within the text. One of my favorites is:

Don’t string adjectives together, especially if they are really nouns. Many high quality pure mathematics original research journal article sentences illustrate this problem.

• 1 $\begingroup$ Thanks for your answer and your paper. The paper seems very helpful for me. I'v downloaded it:) $\endgroup$ –  Paul Commented Aug 11, 2012 at 1:58
• 8 $\begingroup$ +1 for the intelligent sentence. Made me laugh this morning! $\endgroup$ –  Gottfried Helms Commented Aug 12, 2012 at 8:34
• 1 $\begingroup$ it is funny)))) $\endgroup$ –  Seyhmus Güngören Commented Aug 23, 2012 at 8:54

There are some notes on Mathematical Writing from a course taught by Knuth. They are quite extensive; I've only read the first few pages and those were already quite helpful to me, but there are also notes from guest lectures by various people, e.g. Wilf and Halmos.

You say you haven't published a paper yet. Then perhaps this would be useful to you: How to Write Your First Paper by Steven G. Krantz (Notices of the AMS, December 2007, pp. 1507-1511).

I would like to add the above mentioned Paul Halmos work on How to Write Mathematics .

In too short (taken from the MAA ):

Say something. To have something to say is by far the most important ingredient of good exposition. Speak to someone. Ask yourself who it is that you want to reach. Organize. Arrange the material so as to minimize the resistance and maximize the insight of the reader. Use consistent notation. The letters (or symbols) that you use to denote the concepts that you’ll discuss are worthy of thought and careful design. Write in spirals. Write the first section, write the second section, rewrite the first section, rewrite the second section, write the third section, rewrite the first section, rewrite the second section, rewrite the third section, write the fourth section, and so on. ( Annotation : 1,2,1,2,3,1,2,3,4,...) Watch your language. Good English style implies correct grammar, correct choice of words, correct punctuation, and common sense. Be honest. Smooth the reader’s way, anticipating difficulties and forestalling them. Aim for clarity, not pedantry; understanding, not fuss. Remove the irrelevant. Irrelevant assumptions, incorrect emphasis, or even the absence of correct emphasis can wreak havoc. Use words correctly. Think about and use with care the small words of common sense and intuitive logic, and the specifically mathematical words (technical terms) that can have a profound effect on mathematical meaning. Resist symbols. The best notation is no notation; whenever it is possible to avoid the use of a complicated alphabetic apparatus, avoid it.

I would like to add

"If you work eight hours to save five minutes of the reader's time, you have saved over 80 man-hours for each 1000 readers, and your name will be deservedly blessed down the corridors of many mathematics buildings." p. 134

and emphasize that at least the former holds for every scientific writing.

http://padic.mathstat.uottawa.ca/~mnevins/latex/sample.pdf , here is a good short and simple guide. other is the following: www2.kenyon.edu/Depts/Math/Aydin/.../Report.doc, which goes to the main aspects of the body of a math report.

Hope to be useful. Greetings.

• $\begingroup$ www2.kenyon.edu/Depts/Math/Aydin/Teach/Spring03/112/Report.doc $\endgroup$ –  erder Commented Jul 21, 2014 at 2:40

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IB Math EE - Math Extended Essay Guide

The IB Math Extended Essay is a challenging yet rewarding academic endeavor that allows students to explore and delve deeper into various mathematical concepts and techniques. This article aims to provide valuable insights and guidance on how to successfully navigate through the process of writing an IB Math Extended Essay.

With a focus on conducting effective research, applying mathematical concepts, developing a coherent argument, and managing time and organization, this article offers practical tips to help students overcome challenges and roadblocks they may encounter along the way.

Furthermore, this article emphasizes the importance of showcasing personal engagement in the essay by demonstrating genuine curiosity and passion for the chosen topic. By doing so, students can celebrate their accomplishment in producing a well-written piece that reflects their dedication and understanding of mathematics.

Whether you are just beginning your journey or already immersed in your IB Math Extended Essay, this informative guide will equip you with essential tools to excel in your academic pursuit while ensuring safety within an objective academic writing style.

Key Takeaways

• IB Math Extended Essay is a challenging academic endeavor that allows students to explore mathematical concepts and techniques.
• The essay requires effective research, application of mathematical concepts, coherent argument, and time management.
• Personal engagement and passion are important for a successful extended essay.
• Effective research strategies include consulting scholarly articles, books, and reputable online resources, and following clear plans and systematic methodologies.

Understanding the IB Math Extended Essay

The IB Math Extended Essay is a research project that requires students to investigate a mathematical topic of their choice in depth, demonstrating their understanding and application of mathematical principles. It serves as an opportunity for students to engage in independent research and explore the subject matter beyond the scope of the standard math curriculum. The essay allows students to delve into complex mathematical concepts, develop logical arguments, and present their findings in a structured manner.

To ensure safety within this academic pursuit, it is crucial for students to adhere to ethical guidelines when conducting their research. This includes citing all sources accurately and giving credit where it is due. Additionally, students should prioritize data integrity by using reliable sources and verifying information before incorporating it into their essays.

In order to conduct effective research for the IB Math Extended Essay, students must employ various strategies such as consulting scholarly articles, books, and reputable online resources. They should also consider seeking guidance from experts or professionals who can provide valuable insights on their chosen topic. By utilizing these resources effectively, students can enhance the quality of their research and ultimately produce a comprehensive essay that contributes meaningfully to the field of mathematics.

Transitioning into the next section on conducting effective research without explicitly stating "step," it is essential for students to approach their investigation with a clear plan and systematic methodology.

Conducting Effective Research

To effectively conduct research for an IB Math extended essay, it is essential to approach the task with an objective and impersonal mindset. This ensures that the research is conducted in a systematic and unbiased manner, allowing for reliable and valid results. Here are four helpful tips to consider when conducting research for your IB Math extended essay:

• Define your research question clearly: Start by identifying a specific problem or topic within the field of mathematics that you want to explore. Clearly define your research question to provide focus and direction throughout the process.
• Gather relevant sources: Conduct a thorough literature review to identify existing theories, models, and studies related to your research question. This will help you build a solid foundation of knowledge and understand previous findings in order to contribute something new to the field.
• Collect and analyze data: Depending on your chosen topic, collect data using appropriate methods such as surveys, experiments, or simulations. Ensure that your data collection process is rigorous and follows ethical guidelines.
• Draw meaningful conclusions: Analyze your data using appropriate mathematical techniques and interpret the results objectively. Draw meaningful conclusions that address your research question's objectives while considering any limitations or uncertainties present in the study.

By following these steps, you can conduct effective research for your IB Math extended essay that contributes valuable insights into the field of mathematics. Moving forward into applying mathematical concepts and techniques…

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Applying Mathematical Concepts and Techniques

Applying mathematical concepts and techniques involves seamlessly integrating complex equations, formulas, and algorithms into practical real-world scenarios, unlocking the potential for innovative solutions and transformative insights. By harnessing the power of mathematics, researchers can analyze data, model systems, and make predictions with precision and accuracy. Mathematical tools provide a structured framework to explore relationships between variables, identify patterns, and test hypotheses.

To illustrate the significance of applying mathematical concepts in research, let us consider a 2-column table:

Mathematical ConceptPractical ApplicationProbability theoryAssessing risks in safety engineeringOptimizationOptimizing resource allocation in disaster managementStatistical analysisAnalyzing crime patterns for effective law enforcement

This table demonstrates how mathematical concepts can be applied to enhance safety measures. Probability theory enables engineers to assess risks associated with different design choices. Optimization techniques help allocate limited resources efficiently during emergencies. Statistical analysis allows law enforcement agencies to identify crime hotspots for targeted interventions.

By incorporating mathematical concepts and techniques into research methodologies, researchers can derive meaningful insights that contribute to safe and reliable solutions. In the subsequent section about 'writing a strong introduction,' we will explore how an effective opening sets the stage for further investigation without being overly prescriptive or formulaic.

Writing a Strong Introduction

Engaging the reader and providing context is crucial in writing a strong introduction for an essay. By presenting relevant background information or posing thought-provoking questions, the writer can capture the reader's attention and create a sense of relevance to the topic. Additionally, formulating a clear thesis statement in the introductory paragraph helps establish the main argument of the essay and provides a roadmap for what will be discussed. Lastly, outlining the structure and scope of the essay allows readers to understand how different sections will be organized and what specific aspects will be explored, ensuring clarity and coherence throughout the paper.

Engaging the reader and providing context

In order to fully immerse the reader into the world of mathematics, it is crucial to provide a rich contextual backdrop that allows for a clearer understanding and appreciation of the subject matter. Mathematics can often seem abstract and disconnected from everyday life, leading some individuals to view it as intimidating or irrelevant. By engaging the reader through relatable examples and real-world applications, we can create a safe and approachable environment for learning. Exploring how mathematical concepts are applied in various fields such as engineering, finance, or computer science not only helps students see the practicality of mathematics but also fosters an interest in further exploration. Moreover, providing historical context can highlight the development and significance of different branches within mathematics. This sets the stage for formulating a clear thesis statement that builds upon this foundation of knowledge.

Formulating a clear thesis statement

To effectively convey the main focus of a mathematical exploration, it is crucial to craft a concise and clear thesis statement that serves as a guiding framework for the subsequent analysis. A well-formulated thesis statement not only states the main argument or hypothesis but also provides a roadmap for the reader to follow throughout the essay. In an academic style of writing, it is important to maintain objectivity and impersonality, eliminating personal pronouns to create a sense of professionalism and neutrality. By presenting the thesis statement in an objective manner, the writer can engage with an audience that desires safety and assurance in their understanding of complex mathematical concepts. With a solid thesis statement in place, the subsequent section will outline the structure and scope of the essay seamlessly, ensuring clarity and coherence in presenting mathematical ideas.

Outlining the structure and scope of the essay

In the previous subtopic, we discussed the importance of formulating a clear thesis statement for an IB Math Extended Essay. Now, let's move on to outlining the structure and scope of the essay. This step is crucial as it helps provide a roadmap for the reader to navigate through your work. By clearly outlining the structure and scope, you ensure that all relevant aspects are covered while maintaining coherence and logical flow.

To achieve this, consider incorporating an unordered 3 item bullet list in markdown format:

• Begin with an introduction that introduces the topic and provides context.
• Divide your essay into sections or chapters based on different aspects or arguments.
• Conclude with a summary of key findings and a reflection on their significance.

By following these guidelines, you can create a well-structured essay that effectively communicates your ideas. Transitioning into the next section about developing a coherent argument, it is essential to build upon this foundation by presenting compelling evidence and reasoning to support your thesis statement.

Developing a Coherent Argument

Developing a coherent argument requires the use of logical reasoning and well-structured evidence. When constructing an argument, it is essential to present facts and information in a clear and organized manner. This involves avoiding personal biases and emotions, as well as eliminating personal pronouns to maintain objectivity.

To ensure a strong argument, it is crucial to rely on credible sources and data that support the claims being made. Using evidence from reputable academic journals or textbooks helps establish credibility and adds weight to the argument being presented. Additionally, incorporating logical reasoning allows for connections to be made between different pieces of evidence, further strengthening the overall coherence of the argument.

When writing for an audience that desires safety, it is important to use language that is appropriate and sensitive. Avoiding inflammatory or offensive language ensures that readers feel respected and engaged with the content. By presenting arguments in a respectful manner, readers are more likely to consider different perspectives without feeling attacked or threatened.

Developing a coherent argument requires logical reasoning backed by well-structured evidence. By employing these strategies, writers can effectively communicate their ideas while maintaining objectivity. In the subsequent section about using clear and concise language, we will explore how this aspect contributes to the overall coherence of an essay's argumentation process.

Using Clear and Concise Language

Using precise and concise language enhances the clarity and impact of an argument, captivating readers and facilitating their understanding. When writing academically, it is important to eliminate personal pronouns and adopt an objective and impersonal tone. This approach allows the focus to remain on the content rather than the writer's subjective perspective. By using clear and concise language, writers can effectively convey complex ideas without overwhelming or confusing their audience.

In an academic setting, it is crucial to cater to an audience that values safety in their reading material. Using language appropriate for this type of audience ensures that readers feel secure in engaging with the content. It also helps build trust between the writer and the reader, as clear communication fosters a sense of reliability.

Transitioning into the subsequent section about formatting and citations, it is essential to maintain precision throughout all aspects of academic writing. Formatting guidelines ensure consistency in presenting information while citations acknowledge sources used in support of arguments. These elements contribute to establishing credibility within academia by demonstrating a thorough understanding of scholarly norms.

By adhering to these rules of using clear and concise language, as well as following proper formatting guidelines and including accurate citations, writers can effectively communicate their ideas while maintaining professionalism within academic discourse.

Formatting and Citations

This paragraph will discuss the importance of following the IB guidelines for formatting and citation style, using proper citations and references, and creating a bibliography or works cited page. Following the IB guidelines is crucial to ensure consistency and professionalism in academic writing. Proper citations and references are essential to give credit to the original authors and avoid plagiarism. Creating a bibliography or works cited page helps readers locate and verify the sources used in research.

Following the IB guidelines for formatting and citation style

To adhere to the IB guidelines for formatting and citation style, it is essential to employ an academic writing style that maintains objectivity and avoids personal pronouns, thus conveying a sense of professionalism and scholarly rigor to the intended readership. One effective way to enhance the readability of academic writing is through the use of tables. Tables provide a clear and organized format for presenting data or information in a concise manner. They allow readers to quickly grasp key points and make comparisons between different elements. In this context, a 2 column by 4 row table can be utilized, providing a visually appealing layout that facilitates understanding. By incorporating tables into academic writing, authors can effectively communicate complex ideas while maintaining clarity and coherence.

Moving forward, it is important to also consider the importance of using proper citations and references in order to acknowledge the sources of information used in academic writing without plagiarizing.

Using proper citations and references

A crucial aspect of scholarly writing involves the integration of accurate citations and references, which serve to acknowledge the sources of information employed in academic discourse while maintaining integrity and avoiding plagiarism. By providing proper citations, researchers ensure that their work is transparent and reliable, allowing readers to verify claims and build upon established knowledge. Additionally, citations demonstrate respect for intellectual property rights and ethical research practices. In an academic setting where safety is a priority, precise referencing becomes even more important as it enables readers to access relevant sources and evaluate the reliability of information provided. Moving forward into the next section about creating a bibliography or works cited page, attention to citation details will continue to play a pivotal role in upholding academic standards and fostering a culture of accountability in research endeavors.

Creating a bibliography or works cited page

The creation of a bibliography or works cited page is an essential step in scholarly writing, as it provides a comprehensive list of all the sources referenced in an academic work, enabling readers to access and evaluate the reliability of the information presented. By including a bibliography, authors demonstrate transparency and accountability by acknowledging the contributions of other scholars and researchers. This not only strengthens their own arguments but also allows readers to verify the credibility of the sources used. A well-constructed bibliography typically includes information such as author names, publication dates, titles, and any relevant page numbers. It is important to adhere to specific citation styles, such as APA or MLA, to ensure consistency and accuracy throughout the document. With a complete bibliography in place, authors can confidently move on to seeking feedback and revisions for their work.

Seeking Feedback and Revisions

Revising and seeking feedback on the IB Math EE is an essential step in ensuring a well-rounded and academically rigorous research project. The process of seeking feedback allows for constructive criticism, which can help identify areas for improvement and strengthen the overall quality of the Extended Essay. Here are three key reasons why seeking feedback and revisions are crucial:

• Enhancing clarity: Feedback from teachers, peers, or subject experts can provide valuable insights into areas that may be unclear or confusing. This allows the researcher to refine their arguments and ensure that their ideas are effectively communicated.
• Identifying errors: Through feedback, researchers can identify any mistakes or inaccuracies in their work. This could include mathematical errors, incorrect data analysis techniques, or flawed reasoning. Addressing these issues helps maintain the integrity and accuracy of the research.
• Strengthening analysis: Seeking feedback enables researchers to evaluate whether their analysis is comprehensive and thorough enough to support their claims. It encourages critical thinking by challenging assumptions and highlighting potential gaps in reasoning.

By actively seeking feedback and making necessary revisions based on this input, researchers can produce a more polished and robust IB Math EE. Transitioning into the subsequent section on time management and organization, careful planning is also crucial for successfully completing this project within given timelines.

Time Management and Organization

Efficient time management and effective organization are crucial in successfully completing the IB Math EE within designated timelines. The process of researching, analyzing data, and writing a comprehensive extended essay requires careful planning and allocation of time. By creating a detailed schedule that outlines specific tasks and deadlines, students can ensure that they stay on track throughout the research and writing process.

To effectively manage their time, students should prioritize their tasks based on urgency and importance. Breaking down the project into smaller, manageable chunks allows for a more systematic approach to research and writing. Setting realistic goals for each stage of the EE helps to prevent procrastination and ensures progress is made consistently.

Furthermore, maintaining an organized workspace is paramount to staying focused and minimizing distractions. Keeping research materials, notes, and drafts neatly organized not only saves time but also aids in efficiently retrieving information when needed.

By adhering to these principles of time management and organization, students can confidently navigate through the various stages of their IB Math EE while meeting deadlines. In the upcoming section about dealing with challenges and roadblocks, we will explore strategies for overcoming obstacles that may arise during this demanding process.

Dealing with Challenges and Roadblocks

This paragraph introduces a discussion on dealing with challenges and roadblocks in the context of writing an IB Math EE. It highlights three key points: overcoming writer's block and lack of motivation, managing stress and pressure, and seeking help and support when needed. These issues are important to address in order to maintain productivity and achieve success in the research process.

Overcoming writer's block and lack of motivation

Overcoming writer's block and lack of motivation can be a significant challenge when working on an IB Math Extended Essay. To help navigate these obstacles, it is important to implement strategies that promote productivity and maintain focus. Here are five effective approaches:

• Break the task into smaller, manageable parts.
• Establish a routine that includes dedicated work sessions.
• Seek support from teachers, peers, or online communities.
• Engage in activities that promote relaxation and reduce stress.
• Set realistic goals and celebrate achievements along the way.

Implementing these strategies can create a sense of structure and purpose, which can help overcome writer's block and lack of motivation. By incorporating these practices into your writing process, you can increase productivity while maintaining a healthy balance. Transitioning into managing stress and pressure requires acknowledging the importance of self-care and adopting coping mechanisms to ensure overall well-being.

Managing stress and pressure

Having discussed strategies to overcome writer's block and lack of motivation, it is crucial to address the issue of managing stress and pressure in order to maintain a healthy state of mind during the IB Math Extended Essay (EE) process. The demanding nature of this task can often lead students to feel overwhelmed and stressed, affecting their ability to perform well. It is important for students to develop effective stress management techniques such as time management, relaxation exercises, and seeking support from friends and family. Additionally, engaging in physical activities or hobbies can provide a much-needed break from academic pressures. By learning how to manage stress effectively, students can maintain focus and productivity throughout their EE journey. However, there may be instances when additional help and support are required. Thus, it becomes imperative to understand the significance of seeking assistance when needed while navigating through this challenging task without compromising mental well-being.

Seeking help and support when needed

To ensure a successful completion of the IB Math Extended Essay, it is important for students to recognize the value of seeking assistance and support when facing challenges or difficulties. When students seek help, they are not admitting weakness but rather acknowledging that everyone encounters obstacles at some point. Seeking assistance from teachers, peers, or online resources can provide valuable insights and guidance to overcome any hurdles encountered during the research process. Additionally, reaching out for support can alleviate stress and pressure, promoting a healthier mindset and overall well-being. By actively seeking help when needed, students demonstrate maturity and a commitment to their academic growth. This willingness to seek assistance sets them up for success in showcasing their personal engagement with their chosen topic in the subsequent section.

Showcasing Your Personal Engagement

Demonstrating personal engagement is crucial in an IB Math Extended Essay as it showcases the depth of the student's involvement and passion for the chosen topic. By showcasing personal engagement, students highlight their commitment to exploring complex mathematical concepts and applying them to real-world scenarios. This level of involvement not only allows students to develop a deeper understanding of their chosen topic but also demonstrates their ability to think critically and independently.

To showcase personal engagement effectively, students should clearly articulate their motivation for selecting the particular research question or topic. They should provide a comprehensive overview of their thought process, including any challenges they encountered along the way and how they overcame them. Moreover, students should demonstrate a genuine interest in investigating alternative viewpoints or methodologies related to their research question.

In addition, students can exhibit personal engagement by actively seeking out feedback from teachers, peers, and other experts in the field. This shows a willingness to learn from others' perspectives and incorporate constructive criticism into their work.

By demonstrating personal engagement throughout the essay, students not only establish themselves as credible researchers but also contribute valuable insights to the field of mathematics. Celebrating this accomplishment requires reflecting on one's growth throughout the research process while acknowledging that there is always room for further exploration and improvement.

Celebrating Your Accomplishment

Recognizing the hard work and dedication involved in completing an IB Math EE is essential. The process of researching, analyzing data, and writing the essay requires a significant amount of time and effort. Embracing the learning experience gained from undertaking such a substantial project can lead to personal growth and development as a student.

Sharing your knowledge and insights with others in the IB community is not only an act of celebration but also a way to contribute to the collective learning experience. By disseminating your findings, you can inspire others to explore similar topics or provide valuable guidance for those who may be embarking on their own EE journey. Ultimately, celebrating your accomplishment goes beyond personal satisfaction; it becomes an opportunity to engage with others in meaningful discussions and foster intellectual curiosity within the IB community.

Recognizing the hard work and dedication involved

The arduous journey of an IB Math Extended Essay involves hours spent poring over complex mathematical concepts and tirelessly analyzing data sets. To recognize the hard work and dedication involved in completing this task, it is important to acknowledge the following:

• Intellectual rigor: The IB Math Extended Essay requires students to engage in critical thinking, problem-solving, and independent research. It demands a high level of intellectual rigor and analytical skills.
• Time commitment: Students invest a significant amount of time in conducting experiments, collecting data, and writing their essay. This commitment demonstrates their perseverance and determination to excel academically.
• Attention to detail: Success in the IB Math Extended Essay requires meticulous attention to detail. Students must ensure accuracy in calculations, interpretation of results, and adherence to formatting guidelines.

By recognizing the challenging nature of the IB Math Extended Essay process, we can better appreciate the accomplishments achieved by students. Embracing this learning experience allows for further growth and development in mathematical understanding.

Embracing the learning experience

Embracing the learning experience of completing an IB Math Extended Essay involves immersing oneself in intricate mathematical concepts and adopting a mindset of curiosity and continual growth. It requires a willingness to explore complex problems, analyze data, and formulate logical arguments. Throughout the process, students are encouraged to develop a deep understanding of mathematical principles and apply them in real-world contexts. This kind of engagement fosters intellectual rigor and critical thinking skills, preparing students for future academic pursuits. Additionally, embracing the learning experience allows individuals to appreciate the beauty and elegance of mathematics as they uncover connections between different areas of study. By actively seeking out new knowledge and challenging themselves, students can gain confidence in their abilities while nurturing a passion for mathematics. Ultimately, this mindset sets the stage for sharing one's knowledge and insights with others in the IB community, fostering collaboration and mutual growth within this supportive network.

Sharing your knowledge and insights with others in the IB community

Sharing knowledge and insights with others in the IB community can foster a collaborative learning environment that encourages mutual growth and intellectual development. By sharing our expertise, we contribute to the collective understanding of complex concepts and enhance the quality of education within the IB program. One way to facilitate this exchange is through online forums and discussion platforms, where students can seek guidance from their peers or mentors. Engaging in respectful and inclusive conversations allows for diverse perspectives to be heard, fostering an atmosphere of safety and inclusivity. Additionally, active participation in group study sessions or study groups promotes effective communication skills, critical thinking, and problem-solving abilities. Through this process of sharing knowledge, individuals not only reinforce their own understanding but also contribute to the academic success of others within the IB community.

Frequently Asked Questions

Can you provide a step-by-step guide on how to choose a research question for my ib math extended essay.

Choosing a research question for the IB Math Extended Essay involves considering personal interests, access to data, and relevance to mathematics. It is important to formulate a clear and focused question that can be answered using mathematical methods and analysis techniques.

What are some common mistakes to avoid when conducting research for an IB Math Extended Essay?

Common mistakes to avoid when conducting research for an IB Math extended essay include: insufficient planning, lack of clarity in the research question, inadequate data collection methods, ignoring relevant sources, and neglecting to analyze and interpret the results accurately.

Is it necessary to use advanced mathematical concepts and techniques in my IB Math Extended Essay, or can I stick to basic concepts?

Advanced mathematical concepts and techniques are not necessary for an IB Math extended essay. While basic concepts can be sufficient, it is important to demonstrate a thorough understanding and application of those concepts in a rigorous and logical manner.

How can I effectively incorporate real-life examples and applications of mathematics in my IB Math Extended Essay?

To effectively incorporate real-life examples and applications of mathematics in the IB Math Extended Essay, one should choose relevant and relatable topics, gather accurate data, analyze it using appropriate mathematical techniques, and draw conclusions that reflect a practical understanding of the subject matter.

Are there any specific formatting guidelines or citation styles that I should follow when writing my IB Math Extended Essay?

Yes, there are specific formatting guidelines and citation styles that should be followed when writing an IB Math Extended Essay. These guidelines ensure clarity, consistency, and academic integrity in the essay.

The IB Math Extended Essay is a challenging but rewarding task that requires effective research, application of mathematical concepts and techniques, strong introduction, coherent argument development, time management and organization skills. It is important to be prepared for challenges and roadblocks along the way while showcasing personal engagement in the essay. By following these guidelines, students can successfully complete their IB Math EE and celebrate their accomplishment.

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IB Maths EE examples

Filter exemplars, to what extent prime counting function and the riemann zeta function are similar, and what is the importance of their similarities, how are elliptic curves used in cryptography to make information secure in today’s world, want to get full marks for your ee allow us to review it for you 🎯, how does elliptic curve cryptography ensure secure communication of information on the internet, to what extent is the naive bayes classifier effective in classifying iris plants data set, to what extent is the insight into the concepts of consonance and dissonance in music theory provided by the mathematical descriptions of the string motions within stringed instruments described by the fourier series, fast track your coursework with mark schemes moderated by ib examiners. upgrade now 🚀, how can laplace transformation be used to solve differential equations in undamped vibration and kirchhoff's laws of current and voltage, to what length does the lotka-volterra model help us understand the predator-prey relation in an ecosystem, how does the implementation of pell's equation in rsa cryptography affect its efficiency, and what is the security of the prime fake modulus variant against fermat's factorization and wiener's attack, how can the exit velocity vector of a spacecraft performing a hyperbolic flyby manoeuvre be calculated, how do laplace transformations and complex exponential substitution enable the solution of a first order differential equation to obtain the sinusoidal current in a basic rl circuit, the analysis of approaches and extension of a combinatorial geometry problem, to what extent, statistically, are the causal effects of parenting programs on rural chinese toddlers’ cognition different between genders, to what extent is the mathematical proof of ‘ac ohm’s law’ applicable to ac circuits, dans quelle mesure, l’utilisation de pi (π) est-il devenu un facteur important dans la vie quotidienne et l’évolution du monde, how can matrix analysis of game theory be used to guide foreign policy in the ongoing us-iran nuclear tensions, how does the idea of transpositions from group theory apply itself to the solution of the futurama problem, how can the surface area of a violin's top plate be calculated, what is the height and number of towers in a suspension bridge between tuas and jurong island such that it will bear the heaviest possible load and have the lowest cost, how trigonomotry applied in astronomy, to what extent do trigonometry, elliptical geometry and calculus aid forensic analysts in blood spatter analysis and the estimation of the time of death of a corpse, how does the fourier series is used as a musical signal for chord structure, what is modular arithmetic, and how can it be applied to solve problems in the field of number theory, to what extent do the motion of strings within stringed instruments described by fourier series give insight into consonance and dissonance in music theory, how can we use fourier series to analyse and produce an electrocardiogram signal, (35/36) hyperbolic flyby trajectories in orbital mechanicss, analyzing the distribution of cards in the "egg game”, what are some ways the riemann zeta function is connected to the prime numbers, that helps us investigate and better understand them, generating arbitrary uniform probability distributions, to what extent the areas of mathematics such as differ- ential geometry and calculus of variations can be used to generalize the brachis- tochrone problem at planes to curved surfaces embedded in three dimensions, probability factors that influence the game of roulette, we will investigate whether the supply and demand affect the performance of a stock in the us equity market, to what extent does the distance between the path that goes through the gold nucleus and one of the asymptotes of the α particle’s hyperbolic trajectory affect the trajectory of the α particle in rutherford's gold foil experiment, what is the most optimal point on football field to score a goal, how does abstract mathematics apply to physical phenomena in the process of the development of schrodinger’s equations, the rsa algorithm and its vulnerabilities, how can mathematics be used to work out the optimal distance from the try line to position the ball for a conversion kick in rugby union.

Home — Essay Samples — Life — Passion — Why I Love Math

Why I Love Math

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Published: Mar 13, 2024

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The beauty of patterns, the power of problem solving, the language of the universe, the joy of discovery.

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Mathematics Extended Essay Topics for IB

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• Writing Metier

Ever felt the exhilaration that comes with cracking a complex math problem? That’s the sheer beauty of mathematics – it’s a world where numbers dance, patterns emerge, and solutions await the keen observer. Felt that? Hah?

Now, imagine channeling that same enthusiasm and zest into your Math Extended essay. It might sound like a challenge, but isn’t that what we thrive on? With the right Math EE topic, you can showcase your mathematical prowess and leave an indelible mark on your IB evaluators following the general criteria. 

To help you with that, I’ve curated a list of Math extended essay ideas that span the spectrum from the accessible to the intricate. 

So, let’s begin.

IB Math extended essay ideas

These aren’t just topics; they’re gateways to exploration, understanding, and innovation. 

Ready to get started?

Easy-to-approach topics

For those looking for a gentler introduction to the Math Extended essay, below is a list of 10 engaging and manageable IB Math EE topic ideas, each with a research question and a brief description on how to approach it:

Patterns in Pascal’s Triangle

Research Question: 

How do patterns emerge in Pascal’s Triangle?

Description: 

Begin with the construction of Pascal’s Triangle, then delve into the exploration of its patterns, such as triangular numbers and the Fibonacci sequence.

Mathematics of Voting Systems

How do different voting systems weigh individual votes mathematically?

Introduce various voting systems, then analyze the mathematical principles behind each, discussing fairness and potential biases.

Geometry in Art

Research Question:  

How is geometry used to create visual appeal in art?

Explore various art pieces, highlighting the geometric principles employed, and discuss their impact on aesthetics.

The Math Behind Baking

How do ratios in baking recipes affect the final product?

Investigate the importance of maintaining specific ratios in baking recipes and experiment with variations to observe changes in outcomes.

Probability in Card Games

How does understanding probability enhance success in card games like Poker?

Delve into the rules of Poker and then analyze various hands and scenarios using probability principles.

Symmetry in Nature

How is symmetry exhibited in natural formations?

Explore the concept of symmetry mathematically and then identify and analyze its presence in various natural formations like leaves, flowers, and animals.

Mathematics of Music

How do mathematical patterns influence musical scales and rhythms?

Investigate the structure of musical scales, rhythms, and time signatures, highlighting the underlying mathematical patterns.

Simple Interest vs. Compound Interest

How do simple and compound interest methods impact savings over time?

Define both interest methods and then use mathematical modeling to compare their effects on savings over various periods.

The Golden Ratio in Architecture

How is the Golden Ratio applied in iconic architectural designs?

Introduce the concept of the Golden Ratio and then analyze its application in various famous architectural structures.

Mathematics of Juggling

How do mathematical patterns and sequences dictate juggling techniques?

Explore basic juggling techniques and patterns, analyzing the mathematical sequences that jugglers use to maintain rhythm and control.

Each of these topic ideas and RQs is approachable yet offers ample scope for exploration and analysis within the realm of mathematics.

Intermediate topics

For those seeking a bit more challenge without diving into the deep end, the topics below offer a balanced blend of complexity and accessibility.

Here’s a list of 10 average-difficulty IB Math Extended essay topics, each with a research question and a brief description of how to approach it:

Game Theory in Economics

How does game theory provide insights into competitive market behaviors?

Introduce game theory principles and use business world case studies to show strategic decision-making influenced by game theory.

Fractals and Nature

How do fractal patterns manifest in natural phenomena?

Description:  

Delve into the mathematical concept of fractals and explore their appearance in various natural settings, such as coastlines, mountains, and plants.

Mathematics of Population Growth Models

How do mathematical models accurately predict population growth in urban areas?

Introduce population growth models and use real urban area data to validate these models, discussing influencing factors.

Cryptography and Internet Security

How do modern cryptographic techniques ensure data security on the internet?

Explore the evolution of cryptography, focusing on modern techniques, and discuss their role in ensuring data security in online transactions.

Mathematical Modeling of Epidemics

How can mathematical models predict the spread of infectious diseases?

Introduce basic epidemiological models and analyze their effectiveness in predicting disease spread using real-world data.

Topology and Coffee Cups

How does topology explain the mathematical similarity between a coffee cup and a donut?

Dive into the basics of topology and use it to explain the often-cited comparison between coffee cups and donuts, emphasizing the concept of homeomorphism.

Chaos Theory and Weather Forecasting

How does chaos theory impact the accuracy of long-term weather forecasts?

Introduce chaos theory, then delve into its application in meteorology, discussing the challenges and limitations in predicting weather.

Mathematics Behind Neural Networks

How do mathematical algorithms drive the functioning of neural networks in artificial intelligence?

Explore the structure of neural networks and delve into the mathematical algorithms that enable their learning and functioning.

Optimization Problems in Logistics

How can mathematical optimization improve efficiency in supply chain logistics?

Investigate real-world logistics challenges and demonstrate how mathematical optimization techniques can offer solutions.

Number Theory in Modern Cryptography

How does advanced number theory underpin modern cryptographic techniques?

Delve into number theory concepts and their application in modern cryptographic methods, emphasizing their role in ensuring data security.

These topics and research questions offer a balanced blend of complexity and accessibility, making them suitable for students seeking a moderate challenge in their Math Extended essay.

Advanced topics (with potential for higher rewards)

For the brave hearts ready to dive deep, these topics are challenging but can yield impressive results when tackled effectively.

The list of 10 challenging IB Math Extended essay ideas, each with a research question and a brief description:

Navier–Stokes Existence and Smoothness

What are the mathematical implications of the unsolved Navier–Stokes existence and smoothness problem in fluid dynamics?

Delve deep into the Navier–Stokes equations, exploring their significance in fluid dynamics and the challenges surrounding their unsolved problems.

Riemann Hypothesis and Prime Numbers

How does the Riemann Hypothesis relate to the distribution of prime numbers?

Investigate the Riemann Hypothesis, its historical context, and its profound implications for the distribution of prime numbers.

Quantum Cryptography and Key Distribution

How does quantum cryptography enhance security in key distribution?

Explore the principles of quantum mechanics applied to cryptography, focusing on the advantages and challenges of quantum key distribution.

Elliptic Curves and Cryptography

How do elliptic curves underpin modern cryptographic techniques?

Dive into the mathematics of elliptic curves and their pivotal role in modern cryptographic algorithms.

Four Color Theorem and Graph Theory

How does the Four Color Theorem demonstrate the complexities of graph coloring?

Investigate the history and proof of the Four Color Theorem, discussing its implications and challenges in graph theory.

Complex Analysis in Quantum Mechanics

How does complex analysis influence the mathematical foundations of quantum mechanics?

Delve into the principles of complex analysis and explore their applications and significance in quantum mechanics.

Twin Prime Conjecture

What are the implications and challenges of the Twin Prime Conjecture in number theory?

Investigate the Twin Prime Conjecture, its history, and its significance in the realm of number theory.

Knot Theory and DNA Replication

How does knot theory provide insights into the process of DNA replication?

Explore the mathematical intricacies of knot theory and its applications in understanding the complexities of DNA replication.

Ergodic Theory and Thermodynamics

How does ergodic theory connect with the principles of statistical thermodynamics?

Dive deep into ergodic theory, exploring its foundational concepts and its connections to statistical thermodynamics.

Langlands Program and Number Theory

What is the Langlands Program, and how does it seek to unify different areas of mathematics?

Investigate the Langlands Program, its objectives, and its profound implications in unifying disparate areas of mathematics, particularly number theory.

These topics are undoubtedly challenging but can yield impressive results when tackled effectively. They offer a deep dive into complex mathematical concepts and theories, making them ideal for students aiming for the highest grades in their Math Extended essay.

Get extended essay help

Before continuing to the next block, I suggest our Math EE writing or editing services . Our company has been working with numerous IB students and helped them write custom IB extended essays.

Join a team of happy IB students who have followed the IB extended essay rubric and criteria and gained better results with Writing Metier and our Math assignments help !

Overused Math EE topics (proceed with caution)

While these topics are popular, they’ve been explored extensively. If you choose one, ensure you bring a fresh perspective:

• The Fibonacci Sequence in Nature.
• Pi and its Mysteries.
• The Monty Hall Problem.
• The Birthday Paradox.
• Exploring Pascal’s Triangle.
• The Mathematics of the Pyramids.
• The Tower of Hanoi Problem.
• Benford’s Law.
• The Four Color Theorem.
• The Mathematics of the Rubik’s Cube.

Keep in mind that selecting such a topic will require hard work to find a new custom angle of research and sometimes might not lead to higher grades .

In Conclusion

Alright, IB champs, that wraps up our curated selection of Math EE topics, each with its unique flavor and challenge. The topic is just the starting point. The real magic happens when you infuse it with your insights, analysis, and unique perspective. 

Mathematics isn’t just about numbers; it’s about stories and trips within the confines of equations and graphs. As you gear up to tackle your Extended essay, keep that passion alive, keep questioning, and most importantly, enjoy the process. 

Don’t forget to reserve extra space because it may take time to write an extended essay . And if you’re ever in doubt, remember that Writing Metier is here to guide you through every step of the process.

Free topic suggestions

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IB Math Extended Essay Topics: 20+ Ideas for Inspiration

by  Antony W

September 3, 2022

Do you feel stuck on searching for and choosing the best IB Math Extended Essay topics? Or maybe you already have a topic for consideration but it isn’t viable enough to fit the scope of the assignment?

You’ve come to the right place.

In this guide, we’ll give you 15+ IB Math ideas that you can use as inspiration to come up with your own topic or modify and investigate further in your extended essay.

What’s the Purpose of IB Math Extended Essay?

An extended essay in mathematics gives you the chance to demonstrate an understanding of any part of the subject.

You can give an expression on the beauty of mathematics in geometry or fractal theory, the elegance of mathematics in the proving of theorems, and the origin and subsequent development of a branch of mathematics over a period.

The extended essay also allows you to demonstrate the link between the different branches of mathematics and the powerful structures that enable many different problems to be solved by a single theory and the way in which mathematics is applied to real-world situations.

Your essay requires a well-defined and focused research question. An abstract is no longer necessary in an extended essay , but you’re welcome to include it if you believe it will summarizes your strategy to solving the research question and your findings

Math Extended Essay Writing Assistance

While Math opens up an opportunity to explore complex issues beyond the IB Diploma course , the subject can be extremely challenging for some IB students.

If you’re one of the students who chose Math for the Extended Essay project but you find the subject challenging, you can get help online from experts in the subject.

Help for Assessment offers the most  comprehensive Extended Essay help   in Mathematics, even on a topic that you would consider too complex to handle. With our help, you’ll find complex concepts easy to understand, not to mention you’ll get your Math assignment completed on time.

We understand that Math tasks can be costly and involving. However, we’ve made our writing service as affordable as possible, so you can get professional writing help.

We can write you first draft before the first reflection session with your supervisor and equally help you fine-tune the last draft for the final submission.

IB Math Extended Essay Topics

The following are examples of IB Math Extended Essay topics that you might find worth investigating further: 

• What is the return percentage of a certain three-reel slot machine?
• What are the alternatives to Euclidean geometry and what are their practical applications?
• Comparison of population growth models for Country X for the past n years, with projections for the future.
• The exploration of geometric series in musical instruments, such as the fret position on a guitar
• How many convex polygons can you construct from seven tangram pieces?
• An examination of the alteration of a message's truthful substance during transmission between persons
• Is there a connection between the golden ratio and human’s perception on natural beauty, with a focus on the human face and form?
• Is there a relationship between SAT scores and school test results/GPA?
• Leibniz and Newton independently developed calculus at around the same time. Compare their techniques and analyze which notation is currently more prevalent.
• What is the Binomial Theorem, and how has it contributed to human history?
• Complex number problem-solving techniques: What types of real-world issues do complex numbers assist solve?
• Applying Newton's Forward Difference Formula to predict the number of triangles created by subdividing the sides of an equilateral triangle n times.
• RSA Public Key Cryptography's use of modular arithmetic and huge prime integers to achieve anonymity
• Investigating the link between Pascal's Triangle and the Fibonacci sequence
• How can mathematical modeling that employs differential equations determine population growth patterns for a predator and its prey?
• How can the population of cells be determined throughout time? Which mathematical model approximates an actual experiment more precisely?
• A study of Riemann Sums (conventional integration to get areas) and Numerical Integration
• Vedic Mathematics: an investigation of its effectiveness and exploration of its applications
• How are Laplace transforms utilized in the solution of second-order differential equations?
• A statistical examination of the impact of background music on pupils' short-term memory ability
• Analytical and geometric formulations of parabolic and cubic Bezier curves as used in computer graphics software
• A quantitative analysis of the efficacy of two herbs for treating impetigo skin condition
• A study of the nature of beats and the relative consonance of pure-tone dyads
• The link between logical-mathematical intelligence and academic achievement at the undergraduate level
• A study of the link between a bond's coupon rate, yield to maturity, and its clean price
• Using the addition of sine curves to analyze the harmony of Chinese and Western musical scales
• The correlation between students' attitudes toward mathematics and their mathematical performance
• How near is the approximation of the Taylor Series to the original function
• What criteria determine whether the movement of employees on a building site achieves "equilibrium"?
• The efficacy of an English Tuition Program in enhancing the English skills of participants
• A statistical examination of the causes of fatal traffic accidents throughout the holiday season
• Does studying a third language have any influence on the short-term memory retention of lower secondary students?

Keep in mind that this list is not exhaustive by any means. So if none of these topic ideas appeals to you, doing additional research can make a difference.

Related Reading

• The Complete Guide for IB Math Extended Essay
• Get a List of Psychology Extended Essay Topics Here
• Learn More About Physics Extended Essay Assignment

Some Tips to Help You Choose a Relevant Math EE Topic

You can write an extended essay in math on any subject with a mathematical focus. Because IB doesn’t limit the assignment to mathematical theories, you may select mathematical themes from engineering, sciences, and the social sciences.

Statistical analyses of experimental results from other disciplines are also appropriate, as long as they focus on the modeling method and address the limits of the results.

A topic chosen from the history of mathematics may also be acceptable, provided you bring out a clear progression of mathematical growth.

Focusing on the lives of mathematicians or their personal rivalry would be irrelevant and would not score highly on the grading criteria.

Notably, the evaluation criteria provide points for the nature of the inquiry and the extent to which reasoned arguments are suitable for a research issue.

You need to avoid selecting a topic that generates a trivial research question or is insufficiently concentrated to permit suitable presentation in an essay of the right length.

Typically, you will have to challenge yourself to either extend their knowledge beyond the Diploma Program course or apply techniques learned in their mathematics course to the modeling of an adequately chosen topic.

Final Thoughts

One of the most important thing to keep in mind once you find a suitable Math EE topic is that you will be working on an essay, not a research article for a journal of advanced mathematics.

Also, no finding, no matter how remarkable, should appear in your work without proof of your actual understanding of it. 

About the author 

Antony W is a professional writer and coach at Help for Assessment. He spends countless hours every day researching and writing great content filled with expert advice on how to write engaging essays, research papers, and assignments.

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What is an essay in mathematics?

Hi everyone, hopefully it is okay to ask this question here. I have an assignment to write “an essay about some topic in mathematics I find interesting”. I have already chosen the topic (hamming codes), but I am having trouble with what I should actually write. What does an essay in mathematics normally cover?

I’m thinking the reader should be assumed to be unfamiliar with the topic so a part of the essay should probably go to introducing the theory with some short examples, but what should be the “goal” of the essay?

I’m sorry for the vague questions, but the teacher has not given more exact instructions. I’m thankful for any help/advice I can get. :)

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Essays and thoughts on mathematics

Many distinguished mathematicians, at some point of their career, collected their thoughts on mathematics (its aesthetic, purposes, methods, etc .) and on the work of a mathematician in written form.

For instance:

• W. Thurston wrote the lovely essay On proof and progress in mathematics in response to an article by Jaffe and Quinn ; some points made there are also presented in an answer given on MathOverflow ( What's a mathematician to do? ).
• More recently, T. Tao shared some personal thoughts and opinions on what makes "good quality mathematics" in What is good mathematics? .
• G. Hardy wrote the famous little book A Mathematician's Apology , which influenced, at least to some extent, several generations of mathematicians.

Personally, I've been greatly inspired by the two writings listed under (1.) -- they are one of the main reasons why I started studying mathematics -- and, considering that one of them appeared on MathOverflow , I'd like to propose here -- if it is appropriate -- to create a " big-list " of the kind of works described in the above blockquote.

I'd suggest (again, if it is appropriate) to give one title (or link) per answer with a short summary.

• A related question, which I've found very interesting, is Good papers/books/essays about the thought process behind mathematical research .
• Only slightly related (but surely interesting): Which mathematicians have influenced you the most?
• A single paper everyone should read? is not quite related, but still somewhat relevant (especially the most up-voted answer).
• reference-request
• soft-question
• 1 $\begingroup$ Hardy's apology is available here: math.ualberta.ca/~mss/misc/A%20Mathematician%27s%20Apology.pdf $\endgroup$ –  Goldstern Commented Oct 5, 2015 at 15:33
• $\begingroup$ This seems a little broad--can you be a bit more specific? I gave one answer, but do you want things like Dyson's "Birds and Frogs" or Gower's "Two cultures"? $\endgroup$ –  Kimball Commented Oct 5, 2015 at 22:29
• $\begingroup$ @Kimball, first of all, thanks for your answer, the book you suggested seems very interesting. Then, yes, I've read both those articles and, although they didn't come to my mind when I asked the question, they are surely two very insightful additions to this list. Thanks again. :) $\endgroup$ –  user81051 Commented Oct 6, 2015 at 18:12

22 Answers 22

There are many snippets that can be found. I like the following bit of the foreword by Thurston to J. H. Hubbard's Teichmüller Theory . I share the remarks because I think you simply can't have enough of Bill Thurston's insights:

"Mathematics is a paradoxical, elusive subject, with the habit of appearing clear and straightforward, then zooming away and leaving us stranded in a blank haze. Why? It is easy to forget that mathematics is primarily a tool for human thought. Mathematical thought is far better defined and far more logical than everyday thought, and people can be fooled into thinking of mathematics as logical, formal, symbolic reasoning. But this is far from reality. Logic, formalization, and symbols can be very powerful tools for humans to use, but we are actually very poor at purely formal reasoning; computers are far better at formal computation and formal reasoning, but humans are far better mathematicians. The most important thing about mathematics is how it resides in the human brain. Mathematics is not something we sense directly: it lives in our imagination and we sense it only indirectly. The choices of how it flows in our brains are not standard and automatic, and can be very sensitive to cues and context. Our minds depend on many interconnected special-purpose but powerful modules. We allocate everyday tasks to these various modules instinctively and subconsciously. The term `geometry', for instance, refers to a pattern of processing within our brains related to our spatial and visual senses, more than it refers to a separate content area of mathematics. One illustration of this is the concept of correlation between two measurements on a set, which is formally nearly identical with the concept of cosine of the angle between two vectors. The content is almost the same (for correlation, you first project to a hyperplane before measuring the cosine of the angle), but the human psychology is very different. Each mode of thinking has its own power, and ideally, people harness both modes of thought to work together. However, in formalized expositions, this psychological > difference vanishes. In the same way, any idea in mathematics can be thought about in many different ways, with competing advantages. When mathematics is explained, formalized and written down, there is a strong tendency to favor symbolic modes of thought at the expense of everything else, because symbols are easier to write and more standardized than other modes of reasoning. But when mathematics loses its connection to our minds, it dissolves into a haze. I've loved to read all my life. I went to New College of Sarasota, Florida, a small college that was just starting up with a strong emphasis on independent study, so I ended up learning a good deal of mathematics by reading mathematics books. At that time, I prided myself in reading quickly. I was really amazed by my first encounters with serious mathematics textbooks. I was very interested and impressed by the quality of the reasoning, but it was quite hard to stay alert and focused. After a few experiences of reading a few pages only to discover that I really had no idea what I'd just read, I learned to drink lots of coffee, slow way down, and accept that I needed to read these books at 1/10th or 1/50th standard reading speed, pay attention to every single word and backtrack to look up all the obscure numbers of equations and theorems in order to follow the arguments. Even so, when something was ``left to the reader'', I generally left it as well. At the time, I could appreciate that the mathematics was an impressive intellectual edifice, and I could follow the steps of proofs. I assumed that such an elaborate buildup must be leading to a fantastic denouement, which I eagerly awaited -- and waited, and waited. It was only much later, after much of the mathematics I had studied had come alive for me that I came to appreciate how ineffective and denatured the standard ((definition theorem proof)^n remark)^m style is for communicating mathematics. When I reread some of these early texts, I was stunned by how well their formalism and indirection hid the motivation, the intuition and the multiple ways to think about their subjects: they were unwelcoming to the full human mind. John Hubbard approaches mathematics with his whole mind. If you page through the current book, you will see many intriguing figures. That is a first sign: figures are one of the most important ways to keep our thought processes going in our whole brains, rather than settling down into the linguistic, symbol-handling areas. Of course, the figures in your imagination are even more important. Geometric ideas can be conveyed with words and with symbols, sometimes more effectively than with pictures, but a lack of figures is a good indication of a lack of geometry. Another important part of human thinking is the emotional aspect. In mathematics, what is intriguing, puzzling, interesting, surprising, boring, tedious, exciting is crucial; they are not incidental, they shape how we think. Personally, my thinking was shaped by boredom: I develop intense urges to come up with `easy' methods in order to avoid tedious computations that are opaque to me. Hubbard, a principal participant in the mathematics he is discussing, has done an excellent job in conveying the drama."

There are also many very good interviews that can be found, such as this one with Connes , as well as the advice to young mathematicians in the Princeton Companion to Mathematics .

A Mathematician's lament by Paul Lockhart: Reflections on how badly mathematics are taught these days. Imagining how it would be if music was taught the same way.

Indiscrete Thoughts by Gian-Carlo Rota and Discrete Thoughts by Kac, Rota, and Schwartz.

Proofs and Refutations: The Logic of Mathematical Discovery by Imre Lakatos: The sequence of steps through which mathematical ideas can be made to grow in an informal setting is explained through Socratic dialogues between a teacher and students. A beautiful read.

Since you mentioned A Mathematician's Apology : Michael Harris' Mathematics Without Apology .

Here's an excerpt explaining the title:

These attempts at justifications are the 'apologies' of the title. They usually take one of three forms. Pure research in mathematics as in other fields is good because it often leads to useful consequences (Steven Shapin calls this the Golden Goose argument); it is true because it offers a privileged access to certain truths; it is beautiful , an art form. To claim that these virtues are present in mathematics is not wrong, but it sheds little light on what is distinctively mathematical and even less about pure mathematicians' intentions . Intentions lie at the core of this book. I want to give the reader a sense of the mathematical life -- what it feels like to be a mathematician in a society of mathematicians where the first and second lives overlap.

Love and Math: The Heart of Hidden Reality by Edward Frenkel is, in my opinion, a lot better than Lockhart's lament.

The Mathematical Experience by Philip J. Davis and Reuben Hersh is a wonderful collection of essays on mathematics and on the experiences and culture of mathematicians. Written back in the 1980's, it has extremely insightful discussions of many of the same topics that nowadays are discussed on MO. For example, the essay "The Ideal Mathematician," which describes a hypothetical "ideal" mathematician working on the made-up area of "non-Riemannian hypersquares" is absolutely hilarious. Highly recommended!

• 1 $\begingroup$ The "Ideal Mathematician" is, to my mind, a poor mathematician. (It was a caricature, yes, but one which was a little too extreme for me.) $\endgroup$ –  Todd Trimble ♦ Commented Oct 5, 2015 at 16:29
• 1 $\begingroup$ @ToddTrimble, I disliked it too. For myself, the more bearing what I'm working on has on undergraduate or even high-school mathematics, the more excited I am about it. $\endgroup$ –  goblin GONE Commented Aug 23, 2016 at 14:55

Mathematics as Metaphor by Yuri Manin (both the title of the linked book which is a collection of essays, as well as the title of one particular essay in there). At least some of the essays you can find online.

I Want to be a Mathematician , by Paul Halmos.

• $\begingroup$ Indeed I love that book. Thanks for adding it. $\endgroup$ –  user81051 Commented Oct 6, 2015 at 18:13

Eugene Wigner: The Unreasonable Effectiveness of Mathematics in the Natural Sciences

The statement that the laws of nature are written in the language of mathematics was probably made three hundred years ago [It is attributed to Galileo]. It is now more true than ever before … Surely complex numbers are far from natural or simple and they cannot be suggested by physical observations. Furthermore, the use of complex numbers is close to being a necessity in the formulation of the laws of quantum mechanics. It is difficult to avoid the impression that a miracle confronts us here , quite comparable in its striking nature to the miracle that the human mind can string a thousand arguments together without getting itself into contradictions, or to the two miracles of the existence of laws of nature and of the human mind’s capacity to divine them. The closest explanation [for this mathematical universe] is Einstein’s statement that “the only physical theories which we are willing to accept are the beautiful ones” … the concepts of mathematics have this quality of beauty.

• 2 $\begingroup$ I have to disagree. Wigner's assertion that "mathematics is the science of skillful operations with concepts and rules invented just for this purpose" is the whole basis of his piece, and it doesn't have much to do with mathematics. The article is quasi-religious speculation based on this false premise. (The example that Wigner opens the article with is a case in point - he marvels at the appearance of $\sqrt{\pi}$ in the pdf for the normal distribution, as if this were magic. But probability theory was developed with very practical applications in mind.) $\endgroup$ –  Paul Levy Commented May 23, 2017 at 9:31

A Drifter of Dadaist Persuasion by Matilde Marcolli, published in Art in the Life of Mathematicians (Edited by Anna Kepes Szemerédi) American Mathematical Society, 2015, pp.210-231

The Psychology of Invention in the Mathematical Field (Jacques Hadamard's 1945 essay)

• $\begingroup$ This book was very influential to me, and made a huge difference in helping me understand m own process of doing mathematics. $\endgroup$ –  Zach H Commented Jul 17, 2017 at 17:13
• $\begingroup$ I love "the Poincare-Hadamard metaphor" described there! It says that our thoughts conscious and unconscious ones and their interactions could be explained via a mechanical model of states of a system of particles(the details inside). Very inspiring and still I haven't found an enough obstruction to the presented point of view there to the modern neuroscience, but I do not know much about it. An expertise needed! :) $\endgroup$ –  P. Grabowski Commented Apr 14, 2020 at 18:42

The Mathematician by John Von Neumannn.

Enigmas of Chance , by Mark Kac.

I would add "Letters to a Young Mathematician" by Ian Stewart

I recommend:

Vladimir Arnold: Yesterday and Long Ago . This is a very enjoyable and highly interesting collection of anecdotes and historical remarks. The latest Russian edition of this book contains some more chapters. Richard Hamming: You and Your Research , transcribed and edited by J F Kaiser, reprinted in Tveito et al: Simula Research Laboratory . This is the text of a lecture of Hamming.

Birth of a Theorem , by French candidate for Parliament Cédric Villani

• 4 $\begingroup$ Now French member of Parliament Cédric Villani. $\endgroup$ –  Michael Lugo Commented Jul 17, 2017 at 15:16

Here are additional mathematicians' thoughts.

S. Ulam, Adventures of a mathematician .A recollection of his life, from Lwow to Los Alamos. I am linking to excerpts. The book is still available for purchase.

Advices to a Young mathematician , a collection of advice and anecdotes by M. Atiyah, B. Bollobas, A. Connes, D. McDuff and P. Sarnak.

A. Borel, Art and science (Math. Intelligencer vol.5 1983, translation from German). A text for a general audience about the relationship between art and mathematics.

R. P. Langlands Is there beauty in mathematical theories? , this text is actually about number theory, old and new.

T. Gowers The two cultures of mathematics , another take on the dichotomy between problem solving and theory building.

A. Connes A view of mathematics , a thorough exposition of A. Connes'philosophical stance about space and physics. Targeted at a scientific audience.

D. Mumford, the dawning of the age of stochasticity , from algebraic geometry to statistics.

Y. Manin, Interrelations between Mathematics and Physics , on the divergence between mathematics and physics in the XXe century.

M. Gromov, ergobrain , one of the most surprising inquiry about life and mathematics.

I end that list with a text from a french mathematician about the future of mathematics: Poincare, l'avenir des mathematiques .

Perhaps a little broader in range/scope than the original question intended — but then again, perhaps not — the essays collected in

Mathématiques, mathematiciens et société. Publications Mathématiques d'Orsay no. 86 74-16 (1974)

I was led to this when someone somewhere posted a link to Vergne's Témoignage d'une mathématicienne , which is one of the essays in this volume, and — I must confess — is the only one I've read, although the other ones do look interesting

In the Princeton Companion to Mathematics , there is a section entitled Advice to a Young Mathematician (pdf), containing essays by Atiyah, Bollobás, Connes, McDuff and Sarnak.

A Mathematician's Miscellany (reprinted, with additional material, as Littlewood's Miscellany by CUP in 1986) is worthwhile reading.

Clifford Truesdell published a series of essays as An Idiot's Fugitive Essays on Science Methods, Criticism, Training, Circumstances (Springer, 1984), which sets out in a forthright manner the author's views on mathematics and science.

A really nice article by Andrei Toom about mathematical education, especially in the US, got recently mentioned in a comment to this question.

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How Math Autobiographies Build Student Confidence

Incorporating a writing task into lessons can help students reflect on their work and see their skills in a new light.

As educators, we understand the crucial role of fostering confidence and a strong math identity in our students. We aim for each student to feel empowered and to see themselves as capable mathematicians every day. One effective way to nurture this mindset is by encouraging students to write a math autobiography. This reflective exercise invites them to consider their entire mathematical journey, which helps build confidence and reinforce the belief that they are inherently skilled at math.

Writing a math autobiography allows students to connect with each other through their mathematical experiences, identify their learning preferences (though those may change), and set future goals. It also highlights the times when they’ve stepped out of their comfort zone and found joy in the challenges they faced. Take a fifth grader, who discovered that “the more challenging math got, the more I loved it.” This realization helped her embrace challenges, because she knew she could enjoy them. 

Similarly, another fifth grader reflected on his experience learning math during the pandemic. He found that virtual learning strengthened his perseverance, and he realized the power of his own resilience. By integrating reading, writing, and talking, this project guides students to develop a mathematical mindset. 

Would Students Describe Themselves as Mathematicians? 

Most students wouldn’t know how to respond, but we know mathematicians each have their own unique and beautiful math history. We can guide students by helping them find what they enjoy as a mathematician, how they hope to be perceived as a mathematician, and what concepts they’re passionate about. In our experience, students uncovered their math identity by using mentor texts and emotion graphs .

Mentor texts provide a mathematician’s perspective while allowing students to reflect on their own math journey. We chose to share mentor texts centered around the theme of perseverance. When we read Nothing Stopped Sophie , by Cheryl Bardoe, a fourth-grade mathematician stated, ”I forget that every person has trouble with math sometimes.” 

We used emotion graphs to help students share their stories and experiences, which allowed them to realize what has shaped their mindset. This visual brainstorming task was the foundation for their math autobiography. As students thought about their math feelings throughout the years, memories flooded into their heads. 

For our upper elementary students, we decided to use three different emojis on the y axis (happy, indifferent, and sad). The x axis listed the time period of their life. Most students had a “before school” category and then listed each grade level individually. 

Planning on Paper and Identifying Relatable Themes 

With these connections made, students were ready to plan their math autobiographies. Each student got six sticky notes to jot ideas related to their own history. When students read biographies, they look for certain elements (dates, challenges or obstacles, achievements, memorable moments, firsts and lasts, and connections or relationships).

They consider their own math history with each of these lenses, which gives them a collection of ideas to get writing. When students place each sticky note on a sheet of plain white paper, they’re able to view all of their ideas at one time and begin to plan their own autobiography. As they write, they can return to their sticky notes over and over again, weaving in each component. Students might plan ahead by adding a star to certain ideas that they want to include. They might also use check marks to keep track of what they’ve included so far. 

In addition, having the sheet with these sticky notes at student workspaces will help teachers engage in writing conferences to help students get started when they’re stuck (and also help students rehearse ideas with partners before writing).

As students began writing, we noticed that each one crafted their own theme. One student reflected on his confidence in math when he wrote, “When I was in kindergarten, that’s when math started to play a role in my life” and “I realized it was not good to be overconfident in my math skills.”

Another student highlighted her perseverance in her math autobiography, saying, “I knew after trying and working hard, I could do it!” A student chose to focus his writing around connecting mathematics to world experiences when he wrote, “Math, I found, was everywhere—in Sesame Street , on cereal boxes, and in chess, my favorite board game—and so I grew as a mathematician.”

Instructional Strategies That Empower Mathematicians

Having the texts out while students are writing is key. Students quickly pointed out that the mentor texts they were reading had illustrations and pictures, so they included pictures in their own math autobiographies. Students wrote captions to convey the theme of their photographs. 

When describing a candy photo he had pasted on his autobiography, a fifth grader wrote, “Mathematicians sort objects. On Halloween, me and my friends gave ‘values’ to every candy to make trading easier.” Mathematicians became motivated to teach others through their own math story. As a result, they became more confident in their math identity.

Showcase Student Projects in Different Ways 

Some students created storybooks, others designed scrapbook pages, and still others wrote by hand, typed, or recorded their voice. Projects included real photographs, graphs, and math symbols in their design. A gallery walk displayed the math autobiographies so that everyone could move around to see the projects. 

In a follow-up discussion, consider asking students: 

• What’s the same about your work and someone else’s work? What is different? 
• What kind of mathematicians are we? Can you finish this statement, “We’re the kind of mathematicians who…”?
• What can we do with math? How would you finish this sentence, “With math, we can…”? (Do this a few times in different ways.) 

Embedding this writing project into any math classroom can help build confidence in students and promote reflection. Elementary mathematicians will craft a successful autobiography when they have time to reflect on their feelings around math, brainstorm memorable moments, and connect with other mathematicians. 

Reflecting on their current feelings about their identity, the more they reflect, the more they’ll grow in and outside of the classroom.

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181 Mathematics Research Topics From PhD Experts

If you are reading this blog post, it means you are looking for some exceptional math research topics. You want them to be original, unique even. If you manage to find topics like this, you can be sure your professor will give you a top grade (if you write a decent paper, that is). The good news is that you have arrived at just the right place – at the right time. We have just finished updating our list of topics, so you will find plenty of original ideas right on this page. All our topics are 100 percent free to use as you see fit. You can reword them and you don’t need to give us any credit.

And remember: if you need assistance from a professional, don’t hesitate to reach out to us. We are not just the best place for math research topics for high school students; we are also the number one choice for students looking for top-notch research paper writing services.

Our Newest Research Topics in Math

We know you probably want the best and most recent research topics in math. You want your paper to stand out from all the rest. After all, this is the best way to get some bonus points from your professor. On top of this, finding some great topics for your next paper makes it easier for you to write the essay. As long as you know at least something about the topic, you’ll find that writing a great paper or buy phd thesis isn’t as difficult as you previously thought.

So, without further ado, here are the 181 brand new topics for your next math research paper:

Cool Math Topics to Research

Are you looking for some cool math topics to research? We have a list of original topics for your right here. Pick the one you like and start writing now:

• Roll two dice and calculate a probability
• Discuss ancient Greek mathematics
• Is math really important in school?
• Discuss the binomial theorem
• The math behind encryption
• Game theory and its real-life applications
• Analyze the Bernoulli scheme
• What are holomorphic functions and how do they work?
• Describe big numbers
• Solving the Tower of Hanoi problem

Undergraduate Math Research Topics

If you are an undergraduate looking for some research topics for your next math paper, you will surely appreciate our list of interesting undergraduate math research topics:

• Methods to count discrete objects
• The origins of Greek symbols in mathematics
• Methods to solve simultaneous equations
• Real-world applications of the theorem of Pythagoras
• Discuss the limits of diffusion
• Use math to analyze the abortion data in the UK over the last 100 years
• Discuss the Knot theory
• Analyze predictive models (take meteorology as an example)
• In-depth analysis of the Monte Carlo methods for inverse problems
• Squares vs. rectangles (compare and contrast)

Number Theory Topics to Research

Interested in writing about number theory? It is not an easy subject to discuss, we know. However, we are sure you will appreciate these number theory topics:

• Discuss the greatest common divisor
• Explain the extended Euclidean algorithm
• What are RSA numbers?
• Discuss Bézout’s lemma
• In-depth analysis of the square-free polynomial
• Discuss the Stern-Brocot tree
• Analyze Fermat’s little theorem
• What is a discrete logarithm?
• Gauss’s lemma in number theory
• Analyze the Pentagonal number theorem

Math Research Topics for High School

High school students shouldn’t be too worried about their math papers because we have some unique, and quite interesting, math research topics for high school right here:

• Discuss Brun’s constant
• An in-depth look at the Brahmagupta–Fibonacci identity
• What is derivative algebra?
• Describe the Symmetric Boolean function
• Discuss orders of approximation in limits
• Solving Regiomontanus’ angle maximization problem
• What is a Quadratic integral?
• Define and describe complementary angles
• Analyze the incircle and excircles of a triangle
• Analyze the Bolyai–Gerwien theorem in geometry
• Math in our everyday life

Complex Math Topics

If you want to give some complex math topics a try, we have the best examples below. Remember, these topics should only be attempted by students who are proficient in mathematics:

• Mathematics and its appliance in Artificial Intelligence
• Try to solve an unsolved problem in math
• Discuss Kolmogorov’s zero-one law
• What is a discrete random variable?
• Analyze the Hewitt–Savage zero-one law
• What is a transferable belief model?
• Discuss 3 major mathematical theorems
• Describe and analyze the Dempster-Shafer theory
• An in-depth analysis of a continuous stochastic process
• Identify and analyze Gauss-Markov processes

Easy Math Research Paper Topics

Perhaps you don’t want to spend too much time working on your next research paper. Who can blame you? Check out these easy math research paper topics:

• Define the hyperbola
• Do we need to use a calculator during math class?
• The binomial theorem and its real-world applications
• What is a parabola in geometry?
• How do you calculate the slope of a curve?
• Define the Jacobian matrix
• Solving matrix problems effectively
• Why do we need differential equations?
• Should math be mandatory in all schools?
• What is a Hessian matrix?

Logic Topics to Research

We have some interesting logical topics for research papers. These are perfect for students interested in writing about math logic. Pick one right now:

• Discuss the reductio ad absurdum approach
• Discuss Boolean algebra
• What is consistency proof?
• Analyze Trakhtenbrot’s theorem (the finite model theory)
• Discuss the Gödel completeness theorem
• An in-depth analysis of Morley’s categoricity theorem
• How does the Back-and-forth method work?
• Discuss the Ehrenfeucht–Fraïssé game technique
• Discuss Aleph numbers (Aleph-null and Aleph-one)
• Solving the Suslin problem

Algebra Topics for a Research Paper

Would you like to write about an algebra topic? No problem, our seasoned writers have compiled a list of the best algebra topics for a research paper:

• Discuss the differential equation
• Analyze the Jacobson density theorem
• The 4 properties of a binary operation in algebra
• Analyze the unary operator in depth
• Analyze the Abel–Ruffini theorem
• Epimorphisms vs. monomorphisms: compare and contrast
• Discuss the Morita duality in algebraic structures
• Idempotent vs. nilpotent in Ring theory
• Discuss the Artin-Wedderburn theorem
• What is a commutative ring in algebra?
• Analyze and describe the Noetherian ring

Math Education Research Topics

There is nothing wrong with writing about math education, especially if your professor did not give you writing prompts. Here are some very nice math education research topics:

• What are the goals a mathematics professor should have?
• What is math anxiety in the classroom?
• Teaching math in UK schools: the difficulties
• Computer programming or math in high school?
• Is math education in Europe at a high enough level?
• Common Core Standards and their effects on math education
• Culture and math education in Africa
• What is dyscalculia and how does it manifest itself?
• When was algebra first thought in schools?
• Math education in the United States versus the United Kingdom

Computability Theory Topics to Research

Writing about computability theory can be a very interesting adventure. Give it a try! Here are some of our most interesting computability theory topics to research:

• What is a multiplication table?
• Analyze the Scholz conjecture
• Explain exponentiating by squaring
• Analyze the Myhill-Nerode theorem
• What is a tree automaton?
• Compare and contrast the Pushdown automaton and the Büchi automaton
• Discuss the Markov algorithm
• What is a Turing machine?
• Analyze the post correspondence problem
• Discuss the linear speedup theorem
• Discuss the Boolean satisfiability problem

Interesting Math Research Topics

We know you want topics that are interesting and relatively easy to write about. This is why we have a separate list of our most interesting math research topics:

• What is two-element Boolean algebra?
• The life of Gauss
• The life of Isaac Newton
• What is an orthodiagonal quadrilateral?
• Tessellation in Euclidean plane geometry
• Describe a hyperboloid in 3D geometry
• What is a sphericon?
• Discuss the peculiarities of Borel’s paradox
• Analyze the De Finetti theorem in statistics
• What are Martingales?
• The basics of stochastic calculus

Applied Math Research Topics

Interested in writing about applied mathematics? Our team managed to create a list of awesome applied math research topics from scratch for you:

• Discuss Newton’s laws of motion
• Analyze the perpendicular axes rule
• How is a Galilean transformation done?
• The conservation of energy and its applications
• Discuss Liouville’s theorem in Hamiltonian mechanics
• Analyze the quantum field theory
• Discuss the main components of the Lorentz symmetry
• An in-depth look at the uncertainty principle

Geometry Topics for a Research Paper

Geometry can be a very captivating subject, especially when you know plenty about it. Check out our list of geometry topics for a research paper and pick the best one today:

• Most useful trigonometry functions in math
• The life of Archimedes and his achievements
• Trigonometry in computer graphics
• Using Vincenty’s formulae in geodesy
• Define and describe the Heronian tetrahedron
• The math behind the parabolic microphone
• Discuss the Japanese theorem for concyclic polygons
• Analyze Euler’s theorem in geometry

Math Research Topics for Middle School

Yes, even middle school children can write about mathematics. We have some original math research topics for middle school right here:

• Finding critical points in a graph
• The basics of calculus
• What makes a graph ultrahomogeneous?
• How do you calculate the area of different shapes?
• What contributions did Euclid have to the field of mathematics?
• What is Diophantine geometry?
• What makes a graph regular?
• Analyze a full binary tree

Math Research Topics for College Students

As you’ve probably already figured out, college students should pick topics that are a bit more complex. We have some of the best math research topics for college students right here:

• What are extremal problems and how do you solve them?
• Discuss an unsolvable math problem
• How can supercomputers solve complex mathematical problems?
• An in-depth analysis of fractals
• Discuss the Boruvka’s algorithm (related to the minimum spanning tree)
• Discuss the Lorentz–FitzGerald contraction hypothesis in relativity
• An in-depth look at Einstein’s field equation
• The math behind computer vision and object recognition

Calculus Topics for a Research Paper

Let’s face it: calculus is not a very difficult field. So, why don’t you pick one of our excellent calculus topics for a research paper and start writing your essay right away:

• When do we need to apply the L’Hôpital rule?
• Discuss the Leibniz integral rule
• Calculus in ancient Egypt
• Discuss and analyze linear approximations
• The applications of calculus in real life
• The many uses of Stokes’ theorem
• Discuss the Borel regular measure
• An in-depth analysis of Lebesgue’s monotone convergence theorem

Simple Math Research Paper Topics for High School

This is the place where you can find some pretty simple topics if you are a high school student. Check out our simple math research paper topics for high school:

• The life and work of the famous Pierre de Fermat
• What are limits and why are they useful in calculus?
• Explain the concept of congruency
• The life and work of the famous Jakob Bernoulli
• Analyze the rhombicosidodecahedron and its applications
• Calculus and the Egyptian pyramids
• The life and work of the famous Jean d’Alembert
• Discuss the hyperplane arrangement in combinatorial computational geometry
• The smallest enclosing sphere method in combinatorics

Business Math Topics

If you want to surprise your professor, why don’t you write about business math? We have some exceptional topics that nobody has thought about right here:

• Is paying a loan with another loan a good approach?
• Discuss the major causes of a stock market crash
• Best debt amortization methods in the US
• How do bank loans work in the UK?
• Calculating interest rates the easy way
• Discuss the pros and cons of annuities
• Basic business math skills everyone should possess
• Business math in United States schools
• Analyze the discount factor

Probability and Statistics Topics for Research

Probability and statistics are not easy fields. However, you can impress your professor with one of our unique probability and statistics topics for research:

• What is the autoregressive conditional duration?
• Applying the ANOVA method to ranks
• Discuss the practical applications of the Bates distribution
• Explain the principle of maximum entropy
• Discuss Skorokhod’s representation theorem in random variables
• What is the Factorial moment in the Theory of Probability?
• Compare and contrast Cochran’s C test and his Q test
• Analyze the De Moivre-Laplace theorem
• What is a negative probability?

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Essay: Why Is Math Important And How To Pass It?

• Essay: Why Is Math Important…

Why is math important? By now, it is obvious to see that computers are taking our jobs. In a few years, robots will take more than 15% of our current jobs. It teaches us about the importance of math. So why is math important? It’s not too complicated. We use math everyday. For example, we use math when crossing the road, cooking, telling the time, and many more.

It’s possible to live our lives with basic addition, subtraction, and multiplication. Moreover, computers can do more than simple arithmetic. Engineers created the technologies that are taking our jobs. Math is important because if we can’t do what computers can’t do, then we can kiss our jobs goodbye. We can fix it by learning higher math. Students must come together, and we must decide why math is important for our lives.

As college students, we’re expected to have mastered high school math standards. Moreover, some of us had mathematical difficulties in high school.

That’s not a problem because we’re teachable. Brain scan has shown us that the brain can be regrown. It can be regrown through games, food, reading etc. Some of us enrolled in degrees such as nursing, art, math, etc.

All of us in this degree are expected to take math. We all have different reasons for enrolling in math requirement courses, so there’s no need to drop our degrees. It is necessary to understand that everyone has trouble in math, and our teachers had problems in math.

Therefore, we can improve our weaknesses in math by knowing our technological aid, our educational background, and our classroom environments.

Firstly, technological aid are technologies that assist students to learn and help teachers to teach easier. This includes calculators, math multimedia programs, and electronic boards, etc. Mymathlab is a popular technological aid at our school. It’s excellent because it educates students both in class and at home.

It is a sufficient condition because it takes us through the hard problems and provides answers with suitable explanations. According to Abbas Johari, “inductive multimedia programs help students understand math better”. Mr. Johari expanded on a study of multimedia programs using graph and word problems to educate 98 undergraduate students of a large southwest college who were enrolled in a computer literacy course. The research was successful. Because by the end of the research, students showed high improvement in their grades.

The problem is that MyMathLab prevails in spatial learning than in kinesthetic (hands-on learning), auditory (Listening learning), and linguistic(language learning). However, some of us learn differently, and some of us abuse the program because we copy and paste answers. Some researchers believe that students’ performance in math depends on the mathematics curriculum.

According to John K Alsup and Mark J. Sprigler, the ” traditional method(Houghton-Mifflin) showed positive results than the reform method (Cord Applied). The traditional method is based on non-reality problems, but the reform method is based on real problems. It depends on the teachers who teach it. There were about 335 eighth-graders in western United State from different areas that participated in this research. They did it by comparing SAT results of students from both curriculums.

It makes sense because schools such as Harvard University, M.I.T, and UMUC use different math curriculums. Because if schools used the same math curriculum, it wouldn’t make sense to pay high tuition for Harvard. Instead, we can pay less for UMUC and achieve equivalent knowledge.

Mymathlab is part of our curriculum that is given to us by our schools. If we know the program isn’t sufficient, then we should tell the school board. We need to take responsibility for our failures and do better.

Secondly, Educational background refers to our family’s educational values, our learning styles, and our understanding of math. If we know our educational background, then it prepares us for our careers. It is not late to prepare our mindset. Because it reveals our strengths and our weaknesses in math, it gives us the opportunity to grow our weaknesses.

It is a necessary condition that helps us keep track of the lesson. If our parents have a high educational value, then they can help us in math. It helps us know our learning styles. It facilitates learning to the best of our understanding, and it will lead us to do better in math. Knowing all of this can exponentially increase our understanding and performance in mathematics.

Moreover, some of us are doing well without the knowledge of our educational background. So why does it matter?  Some of us have tried it so many times and gave up on it. It is not surprising, math is hard. Some of us have math learning disabilities. The ability of low understanding in math. According to Emmanuel Manalo, Julie K. Bunnell, and Jennifer A. Stillman, “Students with math learning disabilities can improve highly through process mnemonics”.

They conducted a study with 13-14-year old’s, and they were distributed to various groups. Two Experiments were conducted to test Process mnemonics, no instruction, and demonstration imitation. They used different variables. Process mnemonics are using techniques such as rhymes to remember things. Demonstration imitation is the modeling of what you see.

We all know it’s a challenging subject. Some of us have come far without any knowledge of our educational background; If we attain it, then we’re guaranteed to improve our current educational values in math by 5%. Some of us with learning disabilities can improve. According to John Woodward, ” Strategies of teaching facts and extensive practice drills can help develop automaticity in math”.

This study was conducted on 58 fourth graders. Some with learning disabilities, and some with no disabilities. It proves that practice drills and strategies that teachers use to teach us can improve our math understanding.

For example, the speech teacher asked Brandon and his classmates if they knew their learning styles. Since Brandon knew his learning style because his DVR-0061 teacher taught him. He was among the students who knew what she was talking about.  It shows how important learning styles are.

Thirdly, the classroom environment is an important aspect of the understanding. The classroom environment includes climate, teachers, and classmates. Climate is an important fact in understanding math. Because if students are not comfortable in their environment, then they will have a hard time perceiving and processing mathematical information.

For instance, if it is hot outside and a student goes to class. He/she will expect the classroom to be cold. If the temperature in the classroom is the same as the temperature outside, then the student will have a hard time learning. Sometimes, we fail to ask our teachers questions when we don’t understand. It’s important to have a teacher that cares about math.

If teachers don’t care, then students won’t care. We don’t take into consideration that our classmates can tutor us. At times, we understand better from classmates than from our teachers.

According to Karen J. Graham and Francis Fennell, ” Successful teaching depends on teachers’ ability to make decisions based on their knowledge of mathematics, the curriculum expectations, the classroom/ school environment, and the needs of the students”. Our teachers are part of our misunderstanding. If so, why are our teachers allowed to teach us?

Because teachers are meant to teach us depending on our understanding, they are supposed to boost our self-esteem and self-concept in math. Furthermore, teachers are also responsible for everything in the classroom. It implies that teachers need to fix the temperature if it doesn’t correlate with the students in the class.

Is it right to blame our teachers for our failures and misunderstanding? If we think about it as college students, it’s our responsibility to fix our classroom environmental issues. First, we must suggest our opinions with our teachers on class environmental changes. If our teachers can’t help, then we can dress depending on the classroom temperature.

If we don’t understand, then it’s our responsibility to go and find a tutor. For instance, our schools provide us with free tutoring for every subject available. It s up to us to carry ourselves there to learn.

Improvement in mathematical skills is possible through our technological aid, educational background, and classroom environments. Mathematics is important in many ways. First, it is the foundation of the world. Because currently, robots are already taking our jobs, no one will want to employ a person if a computer is faster and better than them. Plus, they don’t need to pay for a computer. Many majors we take today require math, and we need math in our daily lives.

Computers are structured using math, but their math skills are limited. Math is a hard subject, but we can understand it by managing our tech-aides, learning background, and environments. We need to take advantage of the resources around us. Especially us in community colleges because the world does not care about who does the job.

They only want people who can get the job done and fast. We must keep practicing until we understand. Therefore, we must not give up on math under any circumstances because we have dreams to conquer.

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July 1, 2024

The Biggest Problem in Mathematics Is Finally a Step Closer to Being Solved

Number theorists have been trying to prove a conjecture about the distribution of prime numbers for more than 160 years

By Manon Bischoff

Weiquan Lin/Getty Images

The Riemann hypothesis is the most important open question in number theory—if not all of mathematics. It has occupied experts for more than 160 years. And the problem appeared both in mathematician David Hilbert’s groundbreaking speech from 1900 and among the “Millennium Problems” formulated a century later. The person who solves it will win a million-dollar prize.

But the Riemann hypothesis is a tough nut to crack. Despite decades of effort, the interest of many experts and the cash reward, there has been little progress. Now mathematicians Larry Guth of the Massachusetts Institute of Technology and James Maynard of the University of Oxford have posted a sensational new finding on the preprint server arXiv.org. In the paper, “the authors improve a result that seemed insurmountable for more than 50 years,” says number theorist Valentin Blomer of the University of Bonn in Germany.

Other experts agree. The work is “a remarkable breakthrough,” mathematician and Fields Medalist Terence Tao wrote on Mastodon , “though still very far from fully resolving this conjecture.”

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The Riemann hypothesis concerns the basic building blocks of natural numbers: prime numbers, values greater than 1 that are only divisible by 1 and themselves. Examples include 2, 3, 5, 7, 11, 13, and so on.*

Every other number, such as 15, can be clearly broken down into a product of prime numbers: 15 = 3 x 5. The problem is that the prime numbers do not seem to follow a simple pattern and instead appear randomly among the natural numbers. Nineteenth-century German mathematician Bernhard Riemann proposed a way to deal with this peculiarity that explains how prime numbers are distributed on the number line—at least from a statistical point of view.

A Periodic Table for Numbers

Proving this conjecture would provide mathematicians with nothing less than a kind of “periodic table of numbers.” Just as the basic building blocks of matter (such as quarks, electrons and photons) help us to understand the universe and our world, prime numbers also play an important role, not just in number theory but in almost all areas of mathematics.

There are now numerous theorems based on the Riemann conjecture. Proof of this conjecture would prove many other theorems as well—yet another incentive to tackle this stubborn problem.

Interest in prime numbers goes back thousands of years. Euclid proved as early as 300 B.C.E. that there are an infinite number of prime numbers. And although interest in prime numbers persisted, it was not until the 18th century that any further significant findings were made about these basic building blocks.

As a 15-year-old, physicist Carl Friedrich Gauss realized that the number of prime numbers decreases along the number line. His so-called prime number theorem (not proven until 100 years later) states that approximately n / ln( n ) prime numbers appear in the interval from 0 to n . In other words, the prime number theorem offers mathematicians a way of estimating the typical distribution of primes along a chunk of the number line.

The exact number of prime numbers may differ from the estimate given by the theorem, however. For example: According to the prime number theorem, there are approximately 100 / ln(100) ≈ 22 prime numbers in the interval between 1 and 100. But in reality there are 25. There is therefore a deviation of 3. This is where the Riemann hypothesis comes in. This hypothesis gives mathematicians a way to estimate the deviation. More specifically, it states that this deviation cannot become arbitrarily large but instead must scale at most with the square root of n , the length of the interval under consideration.

The Riemann hypothesis therefore does not predict exactly where prime numbers are located but posits that their appearance on the number line follows certain rules. According to the Riemann hypothesis, the density of primes decreases according to the prime number theorem, and the primes are evenly distributed according to this density. This means that there are no large areas in which there are no prime numbers at all, while others are full of them.

You can also imagine this idea by thinking about the distribution of molecules in the air of a room: the overall density on the floor is somewhat higher than on the ceiling, but the particles—following this density distribution—are nonetheless evenly scattered, and there is no vacuum anywhere.

A Strange Connection

Riemann formulated the conjecture named after him in 1859, in a slim, six-page publication (his only contribution to the field of number theory). At first glance, however, his work has little to do with prime numbers.

He dealt with a specific function, the so-called zeta function ζ( s ), an infinitely long sum that adds the reciprocal values of natural numbers that are raised to the power of s :

Even before Riemann’s work, experts knew that such zeta functions are related to prime numbers. Thus, the zeta function can also be expressed as a function of all prime numbers p as follows:

Riemann recognized the full significance of this connection with prime numbers when he used not only real values for s but also complex numbers. These numbers contain both a real part and roots from negative numbers, the so-called imaginary part.

You can imagine complex numbers as a two-dimensional construct. Rather than mark a point on the number line, they instead lie on the plane. The x coordinate corresponds to the real part and the y coordinate to the imaginary part:

Никита Воробьев/Wikimedia

The complex zeta function that Riemann investigated can be visualized as a landscape above the plane. As it turns out, there are certain points amid the mountains and valleys that play an important role in relation to prime numbers. These are the points at which the zeta function becomes zero (so-called zeros), where the landscape sinks to sea level, so to speak.

The colors represent the values of the complex zeta function, with the white dots indicating its zeros.

Jan Homann/Wikimedia

Riemann quickly found that the zeta function has no zeros if the real part is greater than 1. This means that the area of the landscape to the right of the straight line x = 1 never sinks to sea level. The zeros of the zeta function are also known for negative values of the real part. They lie on the real axis at x = –2, –4, –6, and so on. But what really interested Riemann—and all mathematicians since—were the zeros of the zeta function in the “critical strip” between 0 ≤ x ≤ 1.

In the critical strip (dark blue), the Riemann zeta function can have “nontrivial” zeros. The Riemann conjecture states that these are located exclusively on the line x = 1/2 (dashed line).

LoStrangolatore/Wikimedia ( CC BY-SA 3.0 )

Riemann knew that the zeta function has an infinite number of zeros within the critical strip. But interestingly, all appear to lie on the straight line x = 1 / 2 . Thus Riemann hypothesized that all zeros of the zeta function within the critical strip have a real part of x = 1 / 2 . That statement is actually at the crux of understanding the distribution of prime numbers. If correct, then the placement of prime numbers along the number line never deviates too much from the prime number set.

On the Hunt for Zeros

To date, billions and billions of zeta function zeros have now been examined— more than 10 13 of them —and all lie on the straight line x = 1 / 2 .

But that alone is not a valid proof. You would only have to find a single zero that deviates from this scheme to disprove the Riemann hypothesis. Therefore we are looking for a proof that clearly demonstrates that there are no zeros outside x = 1 / 2 in the critical strip.

Thus far, such a proof has been out of reach, so researchers took a different approach. They tried to show that there is, at most, a certain number N of zeros outside this straight line x = 1 / 2 . The hope is to reduce N until N = 0 at some point, thereby proving the Riemann conjecture. Unfortunately, this path also turns out to be extremely difficult. In 1940 mathematician Albert Ingham was able to show that between 0.75 ≤ x ≤ 1 there are at most y 3/5+ c zeros with an imaginary part of at most y , where c is a constant between 0 and 9.

In the following 80 years, this estimation barely improved. The last notable progress came from mathematician Martin Huxley in 1972 . “This has limited us from doing many things in analytic number theory,” Tao wrote in his social media post . For example, if you wanted to apply the prime number theorem to short intervals of the type [ x , x + x θ ], you were limited by Ingham’s estimate to θ > 1 / 6 .

Yet if Riemann’s conjecture is true, then the prime number theorem applies to any interval (or θ = 0), no matter how small (because [ x , x + x θ ] = [ x , x + 1] applies to θ = 0).

Now Maynard, who was awarded the prestigious Fields Medal in 2022 , and Guth have succeeded in significantly improving Ingham’s estimate for the first time. According to their work, the zeta function in the range 0.75 ≤ x ≤ 1 has at most y (13/25)+ c zeros with an imaginary part of at most y . What does that mean exactly? Blomer explains: “The authors show in a quantitative sense that zeros of the Riemann zeta function become rarer the further away they are from the critical straight line. In other words, the worse the possible violations of the Riemann conjecture are, the more rarely they would occur.”

“This propagates to many corresponding improvements in analytic number theory,” Tao wrote . It makes it possible to reduce the size of the intervals for which the prime number theorem applies. The theorem is valid for [ x , x + x 2/15 ], so θ > 1 / 6 = 0.166... becomes θ > 2 ⁄ 15 = 0.133...

For this advance, Maynard and Guth initially used well-known methods from Fourier analysis for their result. These are similar techniques to what is used to break down a sound into its overtones. “The first few steps are standard, and many analytic number theorists, including myself, who have attempted to break the Ingham bound, will recognize them,” Tao explained . From there, however, Maynard and Guth “do a number of clever and unexpected maneuvers,” Tao wrote.

Blomer agrees. “The work provides a whole new set of ideas that—as the authors rightly say—can probably be applied to other problems. From a research point of view, that’s the most decisive contribution of the work,” he says.

So even if Maynard and Guth have not solved Riemann’s conjecture, they have at least provided new food for thought to tackle the 160-year-old puzzle. And who knows—perhaps their efforts hold the key to finally cracking the conjecture.

This article originally appeared in Spektrum der Wissenschaft and was reproduced with permission.

*Editor’s Note (7/9/24): This sentence was edited after posting to better clarify that prime numbers exclude 1.

Understanding Logarithms in Mathematics

This essay about logarithms explores their essential role in mathematics and various applications. Logarithms simplify complex calculations turning multiplication and division into addition and subtraction making them invaluable in scientific research finance and data analysis. Their historical significance is highlighted by contributions from mathematicians like John Napier and Henry Briggs. The essay underscores the importance of logarithms in enhancing computational efficiency and understanding exponential phenomena in natural systems.

How it works

Logarithms abbreviated often so as “register” oneself mathematical contexts serve a critical instrument in areas mathematical and his well-assorted applications. While main embarrasses against much from logarithms on creation put a delegate or activate despite that a base the system is due to be threw open to make the guided room. This concept grows much ??? ????? continues to prop up accounts in areas ranging from finances despite physics.

Gate logarithms substantial part – to simplify accounts include rooms or exponential increase difficult grandes.

Turns an increase and division into appendix and deduction accordingly accounts setting the river logarithms and settle manipulation numeric data easy. It good processing private fanciful person in scientific research where exact accounts substantial part.

In pragmatic terms logarithms find the spacious use in banks in manner from the forests and economy where facilitate them design rhythms increase accounts and culture advantageous signal. Their application draws out he further in statistics and informatics plays in favour of algorithms that analysis power data and courses encoding.

Except that understands logarithms are increased by their understanding exponential positions that describe appearances characterize distribution or disintegration. These terms central in disciplines so as for example dynamic population and radioactive-decay where logarithmic scales assure a show tendencies data light through kind from time.

After their advantage in technical disciplines logarithms too have historical importance. Their imprints reverse display despite mathematicians in manner from John Napier and henry Briggs whose work put foundation for tables that were obligatory instruments for erudite and engineers to appearance logarithmic calculators.

? conclusion while logarithms at a case appear complex at_first splinter their basic role in mathematics and his applications no maybe be overpriced. Liquidates a cut between arithmetic and algebra logarithms simplify accounts and offer penetrating in exponential terms that manage natural systems and does a man. Scope concept logarithms not only increases computational effectiveness and and deepens their understanding principles that bring up our understanding the world mathematics.

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IB Math EE - Propel Your Mind To A 34/34

Unlock the secrets to acing your IB Math EE with a guaranteed 34/34 score! Propel your mind to success and make the most of your potential. 💯

Table of content

Extended essay & what is it❓, ib math ee vs math ia 🥊 , wth is an ib math extended essay 🌋, a brief about ib math ee 🩲, sexy pointers you can't-miss 🤤, 4 ways to find the right math ee topic ⭕, formulate your research question 🔎, planning your research & essay 📖, structuring your essay 🏗️, ib math ee reflection, what to do 🪞, ib math ee criteria 🧮, ib math ee examples 🛹.

It's time we address the elephant in the room.

IB Mathematics Extended Essay.

There, we said it.

If you've taken up IB Math EE, we're here to tell you that it isn't over, contrary to what you might have heard.

Researching, planning, and finally writing a good IB Math EE is no walk in the park, but rest assured that you'll feel different once you've gone through this valuable document.

Nail IB presents the only comprehensive guide you need to nail your IB Math EE.

Let's keep it accurate.

IB Math EE is a rigorous, independent, academic research essay that calls for,

• a keen interest in Mathematics,
• strong mathematical aptitude & reasoning,
• a focused and disciplined approach, and most importantly,
• essential tips, techniques, and quality guidance.

When one opts for IB Math EE, one should ensure a strong passion for Mathematics and a solid mathematical foundation. Not to mention being willing to devote their time and effort to this self-directed research.

Disclaimer -  If you have second thoughts, you should reconsider and re-evaluate your choice because it's no use aiming in the dark.

There needs to be more!

Invaluable tips and techniques are the last missing piece of the IB Math EE puzzle.

Given the panic and stigma around Math Extended Essay, what students need, is not some more panic; instead, expert guidance to keep them pumped to write their EE papers with skill, precision, and utmost zeal. And so, true to our word, we're here to help you.

Say it out loud for me - I got this!

Before diving any further, we are thrilled to share our premium IB Math resources. Check out the  IB Math AI SL  bundle and the  IB Math AI HL  bundle, including everything from comprehensive, up-to-date video lecture courses by Adam Nazha to curated IB Resource Libraries. All your IB DP Math queries are now answered!

While you're at it, check out our  IB Math AA SL  repository, where Patrick Jones (yep, you got that right, our very own  PatrickJMT  - the best Mathematics instructor in the world) will guide you with a brilliant video lecture course he has created to help you nail your concepts and exams alike! Happy learning!

Returning to our one-stop guide for all things IB Math EE, let us see what this article will cover.

• Extended Essay & What is it ❓
• IB Math EE Vs Math IA  🥊
• WTH Is An IB Math Extended Essay?  🌋
• A Brief About IB Math EE ?  🩲
• Sexy Pointers You Can't Miss  🤤
• 4 Ways To Find The Right Math EE topic  ⭕
• Formulate Your Research Question  🔎
• Planning Your Research and Essay  📖
• Structuring Your Essay  🏗️
• Writing Your Essay Writing Your Reflection  🪞
• IB Math EE Criteria  🧮
• IB Math EE Examples  🛹

First things first, let’s briefly discuss what an Extended Essay is.

As the IB defines it, the Extended Essay is a  4000-word in-depth  study, part of the IB Diploma Programme core mandatory for all students.

A  500-word  reflection component-inclusive of 3 reflections - is submitted with the academic writing.

This, of course, is done under the supervision of your IB advisor. The critical thing to note is that you can choose to write an Extended Essay on any subject offered by the IB in your session, though it is highly recommended you settle on a topic you are studying as a part of your IB curriculum.

The purpose of an Extended Essay is to motivate students to conduct formal research on a subject area of interest, independently work on the chosen research question and gain guidance from their respective IB mentors every step of the way.

The IB highlights the significance of the Extended Essay:

• Hands-on learning for undergraduate research
• An opportunity for students to research an area of interest relating to one of the 6 DP subjects opted for by the student.

The skills the Extended Essay aims to develop in students are:

• Coming up with a relevant Research Question
• Personally engaging with a topic of exploration
• A coherent communication of ideas
• Strong Argument building

For better understanding, go through  this document on Extended Essays   by the IB. It will make things more transparent.

Students often confuse their Internal Assessments with Extended Essays, which is a grave mistake. Gaining clarity on both these components of the IB DP core is essential. Let us understand the significant difference between the two.

• While Internal Assessments require students to investigate and explore an idea, Extended Essays are more formal. Students need to perform primary and secondary research on a research question they choose based on a diverse topic about the subject. Since EE is an independently researched essay, your mentor may only be able to give you some of the answers you wanted, unlike the IA.
• Students must write one Extended Essay on a subject of their choice for their IB Diploma Programme, but I must be prepared for all the topics.
• Internal Assessments are graded internally by the subject instructor and then moderated externally. Extended Essays are marked externally by examiners appointed by the IB. Your grade on your Extended Essay is combined with your Theory of Knowledge Essay/Exhibition score to assign you your overall score/points.

Word of caution:  Do not choose the same investigation topic/research question for your Internal Assessment and the Extended Essay.

For a much better breakdown of the EE document, you must go through our article on the  IB Extended Essay . It contains an IB EE checklist you must attend to and a detailed overview of the structure and planning of the EE.

Now that we’ve got our basics to, let’s get down to business. The fear around IB Math Extended Essay makes the most confident students hesitate and doubt their choices. But we aren’t here to scare you. The myth about the horrors of the IB Math EE isn’t all false; it’s a caution. Because unless you’re prepared with ammunition, strategy, and the will to put in the effort, there’s a slim chance you can survive the Math Extended Essay battle.

If you’ve opted for Math Extended Essay, we assume you have a strong passion for Mathematics. When the going gets tough while working on one’s Extended Essay-which, which is usually the case-your only hope at the end of the tunnel is your drive for the subject you’ve chosen. Being a rigorous subject for writing one’s Extended Essay, Mathematics can be incredibly demanding. So unless you want to play it risky, we suggest you reassess your liking for Mathematics. Once you know for sure, that’s when it begins. Lots of research. Since it is an independently written document, you want to be doing your homework well.

We will further break down the IB Math Extended Essay, from planning to execution, so that you can organize your thoughts and make appropriate decisions.

The Math Extended Essay is an in-depth research document covering any topic with a mathematical focus, i.e., not limited to the mathematics theory (shouldn’t be entirely descriptive). IB has divided Math Extended Essays into six categories:

• the applicability of mathematics to solve both real and abstract problems
• The beauty of mathematics examples includes geometry or fractal theory.
• the application of mathematics in the proving of theorems, for example, number theory
• the history of mathematics: the origin and subsequent development of a branch of mathematics over some time, measured in tens, hundreds, or thousands of years
• the effect of technology on mathematics
• in creating links between different branches of mathematics,
• Or in bringing about a new branch of mathematics or causing a particular unit to flourish.

Your Math EE topic should be an extension of your IB Maths syllabus or the practical application of the knowledge you’ve gained in class.

Before moving on to the planning and structure of the Math Extended Essay, here are some pointers you absolutely cannot afford to miss:

• Attempting a Math EE requires a decent amount of hard work. While conducting preliminary research, ensure you land on a topic you can personally engage with and are familiar with. Going for a complicated subject, you can’t comprehend and interpret futile.
• The mathematical work you put in will have to meet a certain level of sophistication, so avoid going for a topic that gives rise to a trivial research question. Your research question shouldn’t be too generic or broad on similar lines. Ensure you can sufficiently narrow down your area of investigation.
• Another thing to remember is that the topic you finally choose should have a good research foundation and room for a novel contribution.
• Incorporate Primary and Secondary research wherever appropriate.
• Proper Referencing should be done for the essay.
• For Reflection, emphasize the Planning, Research, and Writing Processes.

Spitball ideas -  You begin here. Write down any interesting ideas you can come up with. This is your brainstorming session.

Do your Research -  You could not, would not want to meddle with this step, let alone skip it. Research. Read reports, articles, and documentation. Keep jotting important points down simultaneously. The study will and must go on throughout your search for the right math EE topic.

Approach professors/professionals/experts -  Contact concerned lecturers, professors, and industry professionals. They can provide you with invaluable insights. It would be best if you also discussed ideas with your mentor.

Point to Note -  Your goal should be to find a topic you understand and contribute to. It is always recommended to refer to your Math syllabus; something there inspires you to get the ball rolling!

Your essay will be structured as a document answering the Research Question you’re working on. It should be focused, clear and arguable. You will build your essay around your Research Question; hence pay close attention to it.

All you will include in your essay must be mentioned in your Research Question. This means the focused area you will be exploring and the methods you will use to achieve a plausible outcome.

Honing your Research Question is a continuous process. If it appears relatively easy or complex, please rephrase it. Good Research Questions may start with: “How can we model…”, “To what extent…” etc.

Collect Primary and Secondary data and incorporate it wherever appropriate.

Primary sources for Math research include 

• Data-gathering
• Visualization
• Abstraction
• Conjecturing

A few sources to gather Secondary data include Google Scholar, textbooks, databases such as JSTOR, etc. For your essay, know your aims and devise a step-by-step plan to answer your research question while maintaining a mathematical focus.

• Though you can approach your essay in whichever way you like, there should be a consistent logical structure.
• The work done in your essay should develop to answer your research question.
• Subheadings will provide your work with much-needed coherence and viability.
• Describe your way of thinking
• Write definitions and Conjectures.
• Use symbols, theorems, graphs, and diagrams
• Justify your conclusions.
• Visual aids such as graphs and diagrams should be included in the body, while tables and computer programs should be mentioned in the footnotes or the appendix.
• Data must be analyzed appropriately using relevant techniques, correctly reasoned arguments, and adequate methodology to achieve a result.
• Provide the necessary explanation and commentary for all your work.
• Throughout the writing, you may need to re-evaluate the method, approach, and structure to fit the word count adequately.

Once you’re finished writing, you will have to reassess everything from scratch- the equations, the theory, etc., to ensure the coherence of your essay.

• The three reflections should highlight how the process of writing your essay changed in due course of work and what you’ve learned from it.
• The Reflection should emphasize the planning, research, and writing process.
• You should discuss your choices, the re-evaluations you had to make, the limitations of your work, and how you overcame them, if at all.
• The three reflections are written at the beginning, mid, and towards the end of your essay and should have a word limit of approximately 150, 150, and 200, respectively.

Extended Essays are assessed externally by IB-appointed examiners and are marked on a scale of 0 to 34. The score obtained by a student corresponds to a particular band. The bands are:

A -  refers to work of an excellent standard (Level 6)

B -  refers to the creation of a good standard (Level 6)

C -  refers to work of a satisfactory standard (Level-12)

D -  refers to the product of a mediocre standard (Level 4)

E -  refers to the design of an elementary average (Level 6)

Anything below D will be considered a failing grade, and the student will not be awarded the IB diploma.  As we said, your EE is not a component you can mess up.

The Extended Essay Rubric includes the following Assessment Criteria:

Criterion A -   Focus and Method (Covers the topic, research question, and methodology)

Criterion B -  Knowledge and Understanding (Covers context, subject-specific terminology, and concepts)

Criterion C -  Critical Thinking (Covers research, analysis, discussion, and evaluation)

Criterion D -  Presentation (Covers structure, layout)

Criterion E -  Engagement (Covers reflection on planning and progress)

Now that we’ve discovered and learned about the process of writing a Math EE, here are a couple of ideas to give you an idea about how your Math EE questions should be framed:

Topic -  Archimedes’ calculation of areas

Research Question -  What is the legacy of Archimedes’ calculations of circular and parabolic areas in today’s integration methods?

Topic -  Approximation of irrational numbers by rational numbers

Research Question -  How well can π, e, √2, and other irrationals be approximated by rational numbers?

Topic -   The geometry of navigation

Research Question -  What was the role of mathematics and geometry in navigation when we relied on the stars? Does it still play a part now we have manufactured satellites?

You should also check out this  sample band, A Math EE, published by the IB . This will give you an honest insight into making a band-A essay. Also, go through  band B, C, and D extended essays  to understand the loopholes to avoid.

And that’s a wrap!

We hope this comprehensive IB Math EE guide answers your queries regarding the EE. We also suggest you go through  this IB Extended Essay guide  to clarify the Extended Essay component of your IB DP curriculum. You should also try Nail IB’s Extended Essay guide; check it out  here !

At Nail IB, we’re committed to delivering A-quality guidance so that your IB DP journey is smooth and successful. You can explore  our take-test  series here and access   premium guides  here.

Happy learning!

IB Resources you will love!

Essay on My Favourite Subject Maths

Most of the students have a phobia of the subject maths but also it is a beloved subject for many. Although it can be difficult to decide which type of math to take or learn but still it is a very interesting subject. There are a variety of courses to choose from, and it is important to know your learning style and the level of math you want to tackle before making a choice.

It is also important to consider algebra, geometry, calculus, and trigonometry when choosing a math subject. Knowing your strengths, weaknesses, and interests can help make the decision easier. Today we will discuss about my favourite subject maths in detail.

Short and Long My Favourite Subject Maths Essay in English

Here, we are presenting long and short essays on My Favourite Subject Maths in English for students under word limits of 100 – 150 Words, 200 – 250 words, and 500 – 600 words. This topic is useful for students of classes 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12 in English. These provided essays on My Favourite Subject Maths will help you to write effective essays, paragraphs, and speeches on this topic.

My Favourite Subject Maths Essay 10 Lines (100 – 150 Words)

1) Maths is my favourite subject because it is so versatile. It can be used in every aspect of life.

2) I like the challenge of problem solving and understanding the underlying logics.

3) It teaches me to think logically and rationally.

4) Maths gives me the opportunity to explore different perspectives and to develop innovative solutions.

5) The beauty of maths is that it can be applied to so many real-world scenarios.

6) The same mathematical principles can be used to solve problems in many different contexts.

7) Maths has enabled me to develop my thinking skills and to understand how the world works.

8) I love how maths allows me to explore different ways.

9) Maths has become increasingly important in the modern world.

10) One of the best things about maths is that it continues to surprise me with its complexity and application.

Short Essay on My Favourite Subject Maths (250 – 300 Words)

Introduction

Maths is a subject that is essential to our understanding of the world. It has been studied for thousands of years, and its application to many fields, such as science, engineering, economics and finance has revolutionised our understanding of the universe. Maths is all around us, used to interpret and draw conclusions from the problems we face every day. It is an ever-evolving discipline and its influence will only grow stronger in the future.

Why Maths is my Favourite Subject?

Maths has been my favourite subject for many reasons. The most significant factor is that it is a very logical and analytical subject. It requires you to use your logical and analytical skills to solve problems. This helps in improving my problem-solving skills. Moreover, maths is a very wide subject, and as I learn more about it, I understand its application in almost every field. This makes me realise the importance of maths in real life.

Application of Maths

Maths is an integral part of our lives. It is used in almost every field such as engineering, science, finance, and economics. Without maths, none of these fields would be able to function properly. Even in our day-to-day life, we use maths to calculate distances, time, and other measurements.

Maths has been my favourite subject since childhood. Its logical and analytical approach appeals to me the most. Maths is an integral part of our lives and it is used in almost every field such as engineering, science, finance, and economics. It has also helped me to develop certain qualities such as analytical and problem-solving skills, confidence and logical thinking.

Long Essay on My Favourite Subject Maths (500 Words)

Maths is one of the most important subjects for students, it is a universal language of calculations and problem-solving that includes the application of logical thinking. Maths has been and continues to be an essential subject for students as it enables them to understand the world and make sense of concepts that are not easy to understand. Maths also helps them to develop their skills of problem-solving and think logically, which are highly valued by employers.

Maths is a subject with a long history, and it has been studied for thousands of years. Ancient civilizations such as the Egyptians, Greeks, and Chinese all developed their own systems of mathematics. These systems were the basis for modern mathematics, which has been studied and developed over the centuries. Today, mathematics is seen as a vital part of education, and it is an important subject in all levels of schooling.

Importance of Maths

Maths is an important subject because it forms the foundation for many other disciplines. For example, science and engineering rely heavily on mathematics in order to be successful. Mathematics is also important for students to understand the world around them, by helping them to understand how the laws of nature work, or how to solve problems logically.

Maths is also important in everyday life. It gives us an understanding of the world around us, and allows us to make decisions confidently and efficiently. From using numeracy skills to understand basic finances and budgeting, to using more advanced maths to understand complex engineering principles, maths is essential.

Maths can be used to solve a whole range of problems, from simple calculations to complex equations. It can be used to understand concepts such as the law of gravity, or to calculate the probability of a certain outcome. It can also be used to understand the everyday world, such as calculating the cost of an item, or understanding the ingredients in a recipe.

Maths can also be used in a variety of professions. For example, doctors use maths to calculate doses of medicine, and construction workers use maths to calculate and measure materials. In finance, maths is used to analyse risks and predict future trends.

Qualities I Have Developed

Maths has also helped me to develop certain qualities. It has helped me to develop my analytical and problem-solving skills. I have also become more confident in myself as I am able to tackle complex problems. Furthermore, it has also helped me to develop my logical thinking abilities which are essential in many fields.

Maths is a fascinating subject with many applications. Its importance for education and employment cannot be understated, as maths is an essential part of understanding the world around us. Maths also helps problem-solving skills, which are highly valued by employers. For all these reasons, maths should be an important part of every student’s education.

I hope the above-provided essay on My Favourite Subject Maths will be helpful for everyone to know about the subject maths in a better way.

FAQs: Frequently Asked Questions on My Favourite Subject Maths

Ans. The most important concept to learn in maths is understanding the basics and developing problem-solving skills.

Ans. The term is coined during the 6 th century BC by Pythagoreans.

Ans. Archimedes, a Greek mathematician is known as father of mathematics.

Ans. The full form of Math is Mathematics.

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June/July NECO Mathematics Questions 2024 Free Answers (Essay & Obj) 100% Verified Solution

Welcome to NECO 2024 mathematics essay and OBJ answers verified page. See the much awaited 2024 NECO Mathematics Questions June/July legit solution. The examination will take place tomorrow being 10th of July, 2024. The National Examinations Council (NECO) is a Nigerian examination body responsible for conducting and administering national examinations at the secondary school level. NECO plays a crucial role in assessing and certifying students’ academic achievements across various subjects.

In this NECO mathematics exam, students are required to answer questions covering a range of topics including algebra, geometry, trigonometry, and arithmetic. Emphasis will be placed on problem-solving skills, mathematical reasoning, and the application of concepts to real-world scenarios.

When it comes to maths exam, a lot of students are usually afraid. We are hear for you, so you don’t need to be afraid.

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NECO has scheduled the General Mathematics exams for Wednesday, 10th July, 2024. Paper II: Objective will be from 10:00 am to 11:45 am, followed by Paper II: Essay from 12:00 noon to 2:30 pm.

This subject is one of the key subject in the exam, and time is a very big factor as it requires a lot of calculation. Students should manage their time wisely and review key concepts to perform their best.

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MATHS-OBJ 01-10: BDBAEBCCAB 11-20: DBEBABDCDD 21-30: CAEADDDDDE 31-40: BBBEBCDADD 41-50: DDECDEDAEB 51-60: CBCCECCABC

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Mathematicians Are Edging Close to Solving One of the World's 7 Hardest Math Problems

And there’s $1 million at stake.

• In new research, mathematicians have narrowed down one of the biggest outstanding problems in math.
• Huge breakthroughs in math and science are usually the work of many people over many years.
• Seven math problems were given a $1 million bounty each in 2000, and just one has been solved so far.

The “Millennium Problems” are seven infamously intractable math problems laid out in the year 2000 by the prestigious Clay Institute, each with $1 million attached as payment for a solution. They span all areas of math , as the Clay Institute was founded in 1998 to push the entire field forward with financial support for researchers and important breakthroughs.

But the only solved Millennium Problem so far, the Poincare conjecture, illustrates one of the funny pitfalls inherent to offering a large cash prize for math. The winner, Grigori Perelman, refused the Clay prize as well as the prestigious Fields Medal. He withdrew from mathematics and public life in 2006, and even in 2010, he still insisted his contribution was the same as the mathematician whose work laid the foundation on which he built his proof, Richard Hamilton.

Math, all sciences, and arguably all human inquiries are filled with pairs or groups that circle the same finding at the same time until one officially makes the breakthrough. Think about Sir Isaac Newton and Gottfried Leibniz, whose back-and-forth about calculus led to the combined version of the field we still study today. Rosalind Franklin is now mentioned in the same breath as her fellow discoverers of DNA, James Watson and Francis Crick. Even the Bechdel Test for women in media is sometimes called the Bechdel-Wallace Test, because humans are almost always in collaboration.

That’s what makes this new paper so important. Two mathematicians—Larry Guth of the Massachusetts Institute of Technology (MIT) and James Maynard of the University of Oxford—collaborated on the new finding about how certain polynomials are formed and how they reach out into the number line. Maynard is just 37, and won the Fields Medal himself in 2022. Guth, a decade older, has won a number of important prizes with a little less name recognition.

The Riemann hypothesis is not directly related to prime numbers , but it has implications that ripple through number theory in different ways (including with prime numbers). Basically, it deals with where and how the graph of a certain function of complex numbers crosses back and forth across axes. The points where the function crosses an axis is called a “zero,” and the frequency with which those zeroes appear is called the zero density.

In the far reaches of the number line, prime numbers become less and less predictable (in the proverbial sense). They are not, so far, predictable in the literal sense—a fact that is an underpinning of modern encryption , where data is protected by enormous strings of integers made by multiplying enormous prime numbers together. The idea of a periodic table of primes, of any kind of template that could help mathematicians better understand where and how large primes cluster together or not, is a holy grail.

In the new paper, Maynard and Guth focus on a new limitation of Dirichlet polynomials. These are special series of complex numbers that many believe are of the same type as the function involved in the Riemann hypothesis involves. In the paper, they claim they’ve proven that these polynomials have a certain number of large values, or solutions , within a tighter range than before.

In other words, if we knew there might be an estimated three Dirichlet values between 50 and 100 before, now we may know that range to be between 60 and 90 instead. The eye exam just switched a blurry plate for a slightly less blurry one, but we still haven’t found the perfect prescription. “If one knows some more structure about the set of large values of a Dirichlet polynomial, then one can hope to have improved bound,” Maynard and Guth conclude.

No, this is not a final proof of the Riemann hypothesis. But no one is suggesting it is. In advanced math, narrowing things down is also vital. Indeed, even finding out that a promising idea turns out to be wrong can have a lot of value—as it has a number of times in the related Twin Primes Conjecture that still eludes mathematicians.

In a collaboration that has lasted 160 years and counting, mathematicians continue to take each step together and then, hopefully, compare notes.

Caroline Delbert is a writer, avid reader, and contributing editor at Pop Mech. She's also an enthusiast of just about everything. Her favorite topics include nuclear energy, cosmology, math of everyday things, and the philosophy of it all. 

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How the ‘lady gaga of math’ is thinking differently about math education.

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New ideas have the potential to power real progress for students and a new mindset around math ... [+] overall

This post has been updated to include additional context about math education in the United States and globally.

Math education is rapidly changing.

And while there are real headwinds with respect to math achievement , the new ideas and curricula that are emerging have the potential to power real progress for students and a new mindset around math overall. Across the world the scope and sequence of the actual math content being taught is shifting. New research shows that it is part of a growing understanding that the application of math knowledge in the world of work—now, and later—demands different competencies and skills.

Currently, substantive insights are being surfaced and there are critical conversations to be had about math education spanning a wide range of nuances including curricular content, personalization, structuring time, teacher support—and much more. Shifts focused on improving student performance in math are occurring as we speak:

• Korea released an updated curriculum which actually narrows content in elementary and middle school math to allow more time for deep learning.
• Georgia revised its math standards to focus more on strategic math thinking and reasoning to better reflect what is needed in workplaces.
• Florida added data analysis and probability as a core strand of K–5 math.
• Estonia restructured their schedule so that students in Grades 10–12 take an average of 32 short courses per year and must complete a minimum of 96 courses to graduate over three years.
• In Maryland, teachers will have about 40 percent of their working time for planning, collaboration and job-embedded professional learning due the the Blueprint for Maryland’s Future.

Math education for today’s young people should look far different from the traditional mathematics experience that so many of us had, especially when it comes to applying mathematical knowledge to real-world challenges.

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And yet, what happens when a student says, “I don’t like math” or “I’m not good at math”–or even worse, “I’m not doing math”?

The substance of what is being taught in mathematics is truly at the heart of the matter. However, it’s also critically important to build a culture of math confidence that alleviates fears in young people and propels them to value and embrace real-world applications of math.

A passionate voice on math education and how young people experience it is Vanessa Vakharia, who has been called “The Lady Gaga of Math Education”.

Vanessa Vakharia has been called “The Lady Gaga of Math Education”.

A champion and supporter of young people seeing their own math potential come to fruition, Vakharia runs Math Guru , a Toronto-based math tutoring center, hosts a weekly Math Therapy Podcast , has written two Scholastic-published books aimed at helping young people overcome fear and anxiety about math, and has a third book on the way in September.

Our conversation covered a lot of ground including her take on the changes in math instruction, what she sees in her work with young people, what advice she has for educators in helping learners succeed and thrive with math.

Vicki Phillips : Vanessa, I’d like to start with your personal story with math: what was your experience as a student and how does it inform your work today?

Vanessa Vakharia: Once upon a time there was a girl who dreamed of marrying Keanu Reeves and becoming a rockstar … no, but for real though…When I was in high school, I went to a high school in Toronto that was super academic. The message I got every day was: “If you’re not getting an ‘A’, you’re not worth our time. I failed Grade 11 math twice and then squeezed by with a 57% in summer school, at which point, my parents decided it was time to try something new. I ended up at an alternative school down the street.

I walked into my new Grade 12 math classroom the first week of school and looked straight at my math teacher, Ewa (who I’m still best friends with to this day) and said: “You’re going to have a lot of trouble with me, I’M NOT A MATH PERSON.” She looked right back at me and said the words that would forever change my life: “That’s. Not. A. Thing.”

I ended up with 96% in that Grade 12 math class and started wondering what ELSE I could do that I had believed for my entire life that I couldn’t? So I started a rock band , got a commerce degree, then a teaching degree, then decided to study WHY it is that so many women get told they’re not “math people”. I then got a Master’s degree which culminated in a thesis called “Imagining a World Where Paris Hilton Loves Math,” opened Toronto’s first boutique tutoring studio & study lounge ( The Math Guru ), and now travel the world doing Math Therapy with educators—which is essentially professional development targeted toward helping educators heal their students’ math trauma so that they can all build better relationships with math!

VP: Now that is a math story that people need to hear! And it’s a story that is vital at a time when we’re seeing math performance decline around the world and here in the United States. I’d love to hear why you think so many students are struggling with math?

VV: This is a complex question with no one answer. The truth is that students are struggling with math for many reasons, but I’m going to share a thought that we rarely interrogate: the truth is that students have always struggled with math.

If you ask a completely randomized group of 20 adults how they feel about math, I will bet you 10 Canadian Dollars that at least half of them will admit to having a bad relationship with math. Most adults I know today hate math, have avoided certain career paths because they involve math, feel intimidated by anything related to finances or deciphering stats found in the news, and are terrified when their kids bring home their Grade 5 math homework.

We have to ask ourselves what our goal is here. Are we just looking for higher test scores? Or are we looking for a way to raise kids with meaningful, healthy relationships with math that will set them up for a lifetime?

I think it’s time to change the narrative and instead of asking, “ Why are students struggling with math? ” to start asking, “ Why have we been okay with raising kids who have hated math for decades, and how do we finally change that? ”

VP: So true! At the same time, systems can play a role in shaping how students relate to content. A lot of jurisdiction’s new curricula emphasize statistical modeling, data literacy, and career-linked math. Countries like South Korea are narrowing the scope of math content to focus on depth over breadth. What are your thoughts on this approach?

VV: I think that any new curriculum that emphasizes aspects of math that have traditionally not been honored allows students to see that there is so much more to math. It allows everyone to find something mathematical that they can connect with – and that is something that is so instrumental in empowering all students to build better relationships with math!

Depth over breadth is exactly where we should be heading.

If you look at much of our math curricula right now, it has a bit of an everything-but-the-kitchen sink approach at best . We need to ask ourselves what math in school is actually for and redesign our curricula from there. Both students and educators often complain about lack of continuity not only from one grade to another, but from one textbook unit to another.

Yes, the content we teach students needs to be able to take kids to the diverse places they might want to go, but they won’t get there by learning little bits of everything. Instead, by mastering a deep, meaningful relationship with mathematics they can build the knowledge base they need to propel themselves in any direction they might ultimately choose.

VP: That’s a helpful framing of why this shift to depth and more relatable content is potentially so powerful. Another powerful shift is the shift toward personalized learning. As a math educator, and tutor that provides specialized instruction, I’m interested in what you see as the potential benefits and drawbacks?.

VV: Honestly at this point, I think we need to try something different. What we’re doing isn’t working, and it hasn’t been for a long time. I’m all for a new approach to math learning if there’s a chance that we will raise future generations to not only love math, but to feel confident in using mathematics to enrich their lives, both personally and professionally!

It isn’t just what’s best for the individual, but for the whole. We need to raise generations of diverse mathematicians with different skills and unique approaches to solving problems. That is one of the cornerstones of economic, political, and social growth, so it truly serves all of us when we think about math in a way that includes more thinkers.

VP: I’d love to hear what advice you would give to educators and policy makers who are looking to address the math challenge as it were.

VV: My biggest piece of advice would be to give it time.

Too often we implement something new and then panic when it isn’t immediately successful. Meaningful change takes time. Just because something doesn’t work right away doesn’t mean that it won’t ever work.

In fact, any type of change in any type of situation often feels uncomfortable and even disruptive at first, but the real reward is felt once we take the time to adjust, course-correct, and commit to lasting, impactful change instead of settling on a quick fix just to get immediate results. The only way out is through!

VP: Vanessa, I’ve enjoyed this so much. Thank you for your time, your perspective and your passion for helping young people think differently about math!

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Monica Lewinsky: In Praise of Alternate Endings, 10 Years After My First VF Essay

Never lose hope.

“I love you. Bye, Felicia !” I texted my friend Katerina on October 27, 2016. The sassy send-off had been in the culture for two decades (a reference from the film Friday ), but it had only crossed our transom that year. We used it affectionately and, therefore, ironically. Unbeknownst to me, it would be our last text exchange. She died unexpectedly on November 1.

Our friendship had been a salvation in the latter half of what I now call my Dark Decade, roughly 2004 to 2014. Though that stretch of time included some moments of joy, they were few and far between. For the most part, I was in a sea of pain, coming to grips with what it meant to have been standing at the center of a political sex scandal in which I was opposing the most powerful man in the world. Coming to grips with the trauma that grew around me, like weeds, as a result of the public revelations of my private life, the ensuing media circus, an impeachment trial. Coming to grips with what my future might look like. Answer: It looked fucking bleak. I was unemployable. And I was Angry.

Katerina, an entrepreneur and activist, was whip-smart about current events, world history, and spiritual matters. She had a roaring, infectious laugh. She was also kind. You would hardly know that less than a decade earlier, in a freak accident, she had broken her back in five places. After being reassembled with metal rods, she was told she’d likely never walk again. “Screw that,” she would say, “pun intended.” She didn’t lose hope and instead insisted on an alternate ending, prognosis be damned. With grit (and some luck), she recovered and did indeed walk again. And she walked tall.

Our conversations spanned the personal and the political. In 2013, as Edward Snowden leaked classified NSA documents, exposing an array of methods the government and European allies used to spy on private citizens, Kat posited that 15 years earlier, the Starr Report had catapulted us all into what she termed the Age of Transparency. We’d had explosive disclosures in politics before: the Pentagon Papers, Watergate , Iran-Contra. But at their core, these were military, political, professional; 1998 was personal. A boss having an extramarital affair with a young subordinate. A politician abusing power. People, under oath, lying about sex. Rumors titillating the Beltway and beyond. All ordinaire. Almost quotidian. But this time was different. As the truth was made public, published in full on the internet, the personal behavior of a private citizen (me)—along with the actions of others, which had typically been obfuscated by power, gender, status, and wealth—was laid bare. And this transparency led to historical and cultural shifts.

Kat made the point that after 1998, for better or for worse, becoming transparent meant becoming Seen—in new and sometimes disturbing ways. And year upon year, we began to peek behind the veil in all facets of life and culture, thanks to the Patriot Act, reality television, the truth about weapons of mass destruction, the advent of social media, Wikileaks, 23andMe, the UK tabloid phone hacking scandal, and on and on.

Kat’s argument was compelling. And a year after Snowden’s data dump, 2014, I would find myself impacted by this Age of Transparency yet again, this time gratefully.

Ten years ago, after a decade of self-imposed silence in which I had retreated from a world that still shamed me, after a decade of involution and integration (and a fuckton of healing), I jumped back into the public conversation. With no safety net. And I found my voice…by writing for this magazine.

In many ways, my 2014 essay, “ Shame and Survival ,” was a social experiment. Vanity Fair and its then editor, Graydon Carter, could have been lambasted for giving someone from a 15-year-old news story, well, 15 more minutes. And not just in a splashy interview, but in a first-person essay in which I allowed myself to be transparent—and unblinking. (The first line was “How does it feel to be America’s blow job queen?”) I was no longer mediated through another’s gaze but stepping forward unabashedly.

And something surprising happened. A generation that hadn’t lived through the Brainwashing of 1998 insisted on reevaluating this story, one that, given government and judicial overreach, given the technological and tabloid explosion, had always been bigger than me and any one of the other players in it. It had always been a story about the culture at large, and why in my original piece I had referred to myself as a social canvas.

What followed was a Big Fat Fucking Miracle™. My life changed, and I’ll be forever thankful.

That doesn’t mean it has been all smooth sailing. Hardly. Over the years, I’ve spoken confidentially to many people who have been publicly shamed and have explained that taking back one’s narrative doesn’t happen overnight and is (annoyingly) replete with plenty of setbacks. While the essay went on to be nominated for a National Magazine Award (I’m not kidding, alongside pieces by Ta-Nehisi Coates and Roger Angell—Angell won), the following month I attended a party in LA and a famous diva asked me, point-blank, if I was someone’s plus-one. When I replied I had been invited, she snarked, “They just let everyone in tonight, didn’t they?” (Yes, really.)

It went on like this. I gave my first major public speeches: at the Forbes 30 Under 30 conference and, several months later, in 2015, at TED in Vancouver. But mere minutes after my TED Talk, “The Price of Shame ,” went online, the level of vitriol, misogyny, and hatred spewed at me in the comments section was worse than anything the TED team had experienced before. (Who knew there were so many ways to say whore ?) Soon I began working with anti-bullying organizations globally. And yet, when one of the groups was being honored at an event, I was asked not to walk the red carpet.

In 2018, I was asked by Vanity Fair ’s newly appointed editor, Radhika Jones, to address the #MeToo moment in an essay in which I unpacked my own thoughts about what constituted consent in a workplace relationship with a quintessential power differential. Shortly thereafter I was disinvited to a philanthropy summit because former president Bill Clinton was a last-minute addition to the roster.

I could go on for hours. If this last decade has shown me anything, it’s that we never know what lurks or enlivens around the next corner. That one essay, where someone took a chance on me, helped set my life on a different course.

My friend Katerina, who is intensely missed, was not as fortunate. One night she had dined out with her husband and developed what became a fatal case of food poisoning. She was hospitalized, sepsis set in, and she quickly passed. But to this day she remains Seen, in all her common majesty, by everyone who encountered her in life.

After all, in the end what matters more than how it all began is how we have been Seen . And as Rumi wrote:

Never lose hope, dear heart. Miracles swell in the invisible.

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